To my knowledge, each player starts with 20 life in nearly every MTG game ever played. But why exactly was the number 20 chosen? How would the game be different if we started with more or less life? If my friends and I started playing with a higher or lower starting life total, how would that affect our deck construction decisions? Which colors, types of cards, or general strategies would become more or less powerful?
I like shuuja and thesunneversets' answers very much.
There's a theory of Magic that says every matchup is two roles: "beatdown" and "control". The thinking is that, regardless of your actual deck types, one deck's going to have a stronger endgame (putting it in the "control" role), so the other deck has to hurry up and win the game before it's too late (putting it in the "beatdown" role). Raising the life total would naturally make it harder for any deck in the "beatdown" role that wins on damage, however obliquely.
Any deck can conceivably end up in the beatdown role, so don't think that this lets slow decks off the hook entirely. The most obvious disadvantage is to fast aggro decks, because they're more or less playing the beatdown role in every match, and their win condition is, almost universally, dealing 20 damage to the opponent (a few are poison-based instead). Modern-day sets are often balanced under the assumption that a Red-Deck-Wins-style deck should be reasonably competitive but not dominating; this means that the deck's creatures should be able to reliably kill a "goldfish" in around 4 turns (at which point an opponent can drop Day of Judgment to reset the field).
The ability of archetypes like Zoo, Red Deck Wins, and Tempered Steel to deliver fast beatdown sets the pace for the whole environment: other decks play low-cost disruption because not doing so means you get blown out by aggro before you actually get to do your thing. Remove that pressure and you create room for much slower decks. Commander is a great example. With 40 life, you can safely focus on playing high-cost powerful cards and devote your first few turns to hand-sculpting and mana ramp. Mid-priced cards -- strong three-drops and four-drops -- are still very important, but you're likely to see many decks that don't play anything but ramp cards and cheap removal in their low slots. It's possible to build a successful aggro-style Commander deck, but you have to either heavily disrupt the other players' early build-up (so that your dudes have enough time to finish them off while their decks are still in stage 1) or play combo-style elements (e.g. Heartless Hidetsugu) to seal the deal.
In regular 20-life Magic, Skittles is a control finisher. In 40-life EDH, he's an aggro general. That's how much life totals matter!
Similarly, imagine a 10-life game. You go first, play a Goblin Guide, swing for 2. Before I even play my first land I've lost 20% of my life total and I'm facing a four-turn clock. Do you have two Lightning Bolts in your hand? Goblin Grenade and Gut Shot? Fling and Mutagenic Growth? Practically any two or three fast cards will be enough to make me scoop my one land and go home. I have to have some very strong plays on turn 1 (or on your turn 2) in order to even see my own turn 2. What must my deck look like, then, if I'm playing combo and control?
So, in effect, futzing with starting life not only affects the pacing of the game and the relative strength of different decks but also completely changes the window of what's viable in a competitive deck.
Point of order: all sorts of officially endorsed Magic games begin with different starting life totals than 20. Excerpting from the Comprehensive Rules:
118.1. Each player begins the game with a life total of 20.
118.1a In a Two-Headed Giant game, each team begins the game with a shared life total of 30 instead; see rule 810, "Two-Headed Giant Variant."
118.1b In a Vanguard game, each player begins the game with a starting life total of 20, as modified by his or her vanguard card's life modifier. See rule 902, "Vanguard."
118.1c In a Commander game, each player begins the game with a starting life total of 40 instead; see rule 903, "Commander."
118.1d In an Archenemy game, the archenemy begins the game with a starting life total of 40 instead; see rule 904, "Archenemy."
Each of these variants accommodates different number of players, supports different sorts of politics and allegiances, and generally lasts a differing length of time.
I'm unwilling to engage in random speculation about what would happen if you tinker with the life totals in a standard 2-player duel: generally speaking this is a bad idea, because the power level of the cards is balanced quite nicely for 20 point life totals. Lava Axe is a card that is well costed for the fact that it takes away one-quarter of a player's starting life total. If players started with 100 life each, then Lava Axe becomes a considerably more awful card.
Conveniently enough, 20-point life totals usually allow for "best of 3 matches" to be comfortably played inside of an hour in tournament conditions. If you have no time constraints and just want to play Magic all day, there is no reason why you shouldn't bump up the life totals as a house rule. Just be aware that it will completely change the game balance, rendering some cards much better and some cards much, much worse. Working out which is which could be a fun intellectual exercise for the jaded Magic player, but personally I like the fact that a lot of very smart people at Wizards have spent 20 years perfecting the game balance for 20-life-point games... why mess with that?
Not an expert here, but I the obvious change is that it makes it takes more damage to kill a player by damage, making decks that rely on that win condition slower. Then the question is, what sort of decks benefit from this? Clearly not aggro or direct damage, but combo decks would have more time to get their combo out vs one of the more damage-y decks, and control decks would have more time to lock down the board and implement their own win condition.
I think, within the categories of combo and control decks, those deck that win by means other than reducing life total (such as milling decks) will have their speed unaffected, whereas decks that rely on dealing 20 damage will now have to deal 40 damage, so they will be slowed.
As @thesunneversets points out, many individual cards will shift in value, e.g. cards that damage you or have you pay life will have a smaller relative effect on you and thus get better. However cards that deal direct damage to an opponent get relatively worse.
I would oppose your own comment and some of the comments in the other answer: do try it out in a two-player duel! Also try starting with less than 20 life. Find out how to adjust your strategy to these situations. Obviously you shouldn't play a standard game (or build a standard deck) in the same way as you would when starting with 40 or 10 life, but learning those differences will help you understand the game.
20 is a convenient number, as it's the largest number on a 20-sided die, and the largest number comfortably placed around a small spinner; spindown dice have been available since the early days, and I've read that they were part of the original idea. Further, 20 is also the low end of the number of stones in a tube of gaming stones (the high end is about 40), and under half the number of mini-poker-chips in a tube of mini-poker-chips (which is typically 50).
Having played games with 10 LP, it doesn't cut the typical 45-60 minute time in half; it cuts it by about 10-20 minutes, as it takes a bit of time to get damage causing cards into play.
I've not played 30 point, but friends have, and it doesn't really lengthen the game that much, either; they say it adds another 15 minutes or so to the typical 45-60 minute game.