I don't understand why I lost or how territory is defined when both players have a bid for territory. Here I lose by 25 points (5x5 board). But it seems to me that I have an equally valid bid for controlling the left and right halves of the board with my line down the middle. Why does white end up getting the top half and bottom left corner instead?
5x5 boards are actually harder to understand than a little bit larger board. There are a lot of weird situations that can happen because the board is so small, so I would recommend starting with 9x9 instead. This situation isn't too tricky, though.
In this case, the white group on the right is alive. It can make two separate eyes in two different ways - by splitting the area on the right into two, or by surrounding the upper right corner. Black can't stop both, so this group is safe.
The black stone that separates the white stones in the lower left is going to die. Whatever Black does, this stone will never get more liberties than the white stones on the left, so it's going to be captured. Once this happens, all of the white stones are effectively connected and safe.
Since the upper black group doesn't have room to make eyes, it's going to be captured eventually.
The fundamental idea of Go is to get your stones onto the board and keep them there.
The first question you should ask yourself is: what is the greatest number of stones that one player could safely have on this 5x5 board?
Not 25: if every spot were covered, then the stones would not have any liberties, so they would be removed. (And indeed, most people play such that putting the 25th stone of the same colour on the board would be illegal.)
Not 24: if there is only one open spot, then the opponent could play there. Capture of opposing stones is considered first, before the liberties of the stone that was just played. After the opponent captures 24 stones, the stone that was played to do so will have liberties.
But if we leave two open spaces that are not adjacent, then the opponent cannot capture. Trying to place a stone on either intersection would not work (and, again, is normally considered illegal): it cannot immediately capture the opponent's stones, and then it has no liberties for itself.
So our answer is 23. But as you have noticed, on 5x5 we generally consider that the maximum margin of victory (notwithstanding komi) is 25 - i.e., all the points of the board. We could say that there are 25 points "available" to split between the players, and this is the principle of area scoring.
If we kept playing until neither player had a safe move (i.e., every remaining move would allow a capture that is not some kind of tactically sound sacrifice), we might simply count the stones to figure out the winner. This is not exactly right, except for some historical variations, because again we have a tradition of counting every point. But it gives us the right idea. By "territory", we mean the places that are safe moves for us, but not for the opponent. If we play there, our stone will stay indefinitely, assuming proper play afterwards; if the opponent plays there, we can eventually capture it, and the opponent can do nothing to stop us.
With even slightly skilled players, we will normally expect that an equal number of moves are played by each player (possibly one more by Black; some rulesets account for this explicitly) until the end - nobody will pass until there are no more strategic objectives left, because it would obviously be losing to do so. We similarly expect that the game will reach a state where every open space is territory for one player or the other, so it's tedious to put stones on the board to fill space whose "owner" is surely determined (and to capture stones that are powerless to resist). So at this point the players shall agree (if they are using a ruleset with this principle) to stop playing, and count territory and prisoners to find the winner and the margin of victory. This is the principle of territory scoring.
We count the prisoners because they represent stones that the opponent placed, but doesn't have on the board; thus they are a setback from the opponent's tally of already-claimed spaces during the alternating play phase. Similarly, we count "dead" stones that will be inevitably captured, as if they were prisoners; we do not bother with actually capturing them.
To account for the fact that the board can't be completely filled, we typically instead describe territory as space that is surrounded by our "alive" stones that cannot be captured - that is to say, stones such that proper play will protect them from capture, even if we allow the opponent to move first. The idea is:
Stones get captured when their liberties are filled. Therefore, to protect them from capture, they need to be arranged so that the remaining liberties can't be filled. (Liberties, plural, because if there were only one liberty then playing there would obviously capture the stones.)
The most common, but not strictly necessary way of doing that is to have two or more separate, surrounded liberties called eyes, such that neither is a legal move for the opponent. The other ordinary way is to create seki, where an empty space is adjacent to both players' stones, and neither may play there because it allows the other to capture (without subsequently leading to a recapture).
If our stones surround a sufficient amount of space, then we should be able to find a way to split that space apart and make two separate eyes. We can do this at our leisure, only working towards that goal if the opponent strategically forces it. We don't need to make eyes explicitly in order to claim that the stones are alive; we only need to answer any challenge, by doing so before we actually get captured.
If our stones are alive, and also surround some space, then we should be able to fill the space later. If our surrounding wall cannot be captured, then the opponent could only deny those spaces to us by occupying them and then making those stones alive. But this ranges from extraordinarily difficult to utterly impossible - given that we get to play every other move in order to interfere with the attempt.
So, we should reach a point in the game where every stone is known either to be alive or to be dead, and every empty space is either "territory" that is surrounded by a living group or "dame" that is adjacent to both players' stones. (A seki will have such points where neither player wants to play, but there will usually also just be points where either player could play just to fill the space. Usually we will skip filling these, because it's obvious that they will be split equally between the two players.)
The computer has determined that the Black stones are all dead and the White stones are all alive. White wins by 25 points because it is inevitable that, were the players compelled to keep going, White would be able to capture all the Black stones and fill in all the empty points (except for two), while Black would have no way to keep any stones on the board.
I will deliberately explain this without images, because it will be useful for you to train your visualization skill.
The easiest way to understand why this is, is to look at the lower-left first. Black has a single stone on the bottom edge that has only one liberty, and another there with two liberties. White's pair of stones on the left, meanwhile, has three liberties. It's easy to verify that there are no special tricks here; White can capture those stones, even assuming Black gets the next move, and by doing so, the two White stones will be protected - they will join the other White stones and easily make eyes using the space they have surrounded.
This then raises the question about the other three Black stones. Can they not survive? The answer is that they indeed cannot survive. Since White's stones will easily make eyes, there is no chance for seki against them; Black's stones will need to find their own eyes. But there is no way to do this. Even supposing that Black plays the two points above the White pair of stones (useful "forcing moves" that will threaten White and force capturing in the lower left for safety), we can see that the Black stones will simply surround an adjacent pair of spaces in the top left.
And these adjacent spaces cannot make separate eyes. If we require White to demonstrate the capture, it will be easy: after filling the "outside" spaces, White simply plays either of the points inside. If Black captures that stone, it only leaves one space behind, so White can re-capture the entire group. But if Black doesn't capture the stone, White can simply play the other empty space, which is the last liberty of the group. Catch-22.
Exercise: think about what happens if a group surrounds three adjacent empty spaces in a straight line. (For simplicity, assume that every outside space is covered by the opponent's alive stones.) If it is your move, can you play such that the group will have two separate eyes? Where? Is it necessary to do this? (Hint: If it is the opponent's move instead, can the eyes be prevented? How?) Is there the possibility of seki? (Hint: think carefully about what happens if opponent puts two adjacent stones inside this space. You could capture them by playing on the remaining point... but then what?)
Exercise: What about four spaces in a row? What about five spaces? (Hint: use the answer from the three-space case, in order to reason about seki.)
Research problem: investigate surrounded spaces that aren't all in a straight line.
As an adult learner, I strongly advise not to use a 5x5 board. Please do play a game or two on 3x3, because it's essentially the simplest viable option. (I am discounting even-sized boards on principle; but it should be noted that 2x2 is especially weird if you try to rule on what should happen if the players want to continue past the first two obvious moves. And of course, on 1x1 nobody can actually place a stone.) It actually teaches some fundamentals, while being small enough that you should be able to figure out the dominant strategy easily. 5x5 isn't big enough to explain anything to an adult that isn't already understood from 3x3 - in particular, it doesn't really teach about territory, because proper play still leads to a whole-board win for Black. But that's much harder than solving the 3x3 game, and more of an academic exercise.
After making sure you understand what to do on 3x3 (and why it's a clear whole-board win for Black), please move directly to 9x9. This is a much more "standard" board size that you'll be able to find more resources for (unless you're studying game theory at university or something like that). It's large enough that "territory" becomes a meaningful strategic concept.
However, please feel free to play games by trying to fill in all the "safe" intersections, until you feel like you have a proper intuition for territory. Make sure that you can find opponents who are amenable to this - typical rulesets will penalize you, and computer opponents won't be programmed to play along even if they're set to be very weak. This will be easier to do over a real board, assuming you know other beginners. Eventually you will find that this part becomes boring and that you "know what will happen" - that's proof of your understanding.
It would have been clearer to you if you had played on until the boundaries between Black and White were clearly defined. “Territory” means surrounded vacant points.
It was unhelpful to you that the game stopped here, whether that was done by your opponent, a computer or you yourself. The other answers give you the theory, but the easiest way to pick up the theory is by playing your games to the end until you understand what is going on.
But, you may ask, what is “the end”? A fair question, and one that confuses beginners: the end is when both players agree that the ownership of all the spots on the board is settled (which also means the score is clear). In particular, both must agree which stones will end up captured. Until you agree, just go on playing.
The game is finished once you agree which stones cannot escape capture and, if you were to remove all those stones, each spot on the board would:
Once you reach that point, there are two ways of scoring:
Your game was not completely finished because of the neutral spots (between the colours) where it was still possible for at least one player to play. I expect that the computer treated the result as obvious once both sides passed, but really you needed a chance to continue.
The trouble with your “line down the middle” is that it is not connected, and so can be picked off piece by piece.
You have lost because a computer arbiter has decided that all the Black stones are dead.
Therefore White controls the whole board and gets 25 points and Black gets 0 points.
If this had been a human game instead, played face-to-face, then you wouldn't have lost like that. In a human face-to-face game, after both players have passed, White would have asked you: "Do you agree that all your stones are dead?". If you agree, then you lose. If you disagree, then game resumes and you can keep playing until all 'dead' stones have been effectively removed from the board. Then the next time both players pass, all stones that are on the board are considered alive and we can count the points of both players.
Where the computer is not entirely wrong to have marked all your stones dead is that, in this position, presuming White is not a beginner and doesn't make too many mistakes, it is indeed possible for White to capture all Black stones, no matter how well Black plays.
The fact that White can capture all Black stones is due to the very few number of liberties of the two Black stones in the Southwest.
On this diagram you can see that one of the the Black stones has only 2 liberties, which I have marked "1" and "2" in blue:
On this diagram you can see that the group of two White stones has 3 liberties, which I have marked "1", "2" and "3" in red:
Because the White stones have more liberties than the Black stone, White will be able to capture the Black stone. After that, once this Black stone has been captured, the two White stones will be pretty indestructible and will be able to easily capture the three Black stones in the North.
The reason why the computer marked the Black stones as dead without requiring that White actually play out the moves required to capture the Black stones, and the reason why human players often agree that stones are dead even though they haven't been explicitly captured, is because this phase of "effectively play the moves to capture the stones" is considered boring by experienced players, who can see in advance that in this position, the Black stones don't stand a chance and will be captured eventually no matter how well Black plays.
@TimK is right about why Black lost.
How territory is defined -- a player's territory is that part of the board that the player exclusively controls, such that any opponent stones in that area will be inevitably captured because
It's one of those things where it's easier to demonstrate than define. In the game you posted, the whole board is White's territory, because all of the Black stones are inescapably captured (barring some kind of astounding blunder by White).
To develop your sense of the board, I would encourage you to prove to yourself that, with Black and White taking turns as normal, none of the Black stones can live.
One advantage of the 5x5 board is that, from a position like the one shown, it will not take more than a half-hour or so to check every possible line of play. Some hints:
