Determining when to mulligan is a very important skill in magic. But how can I hone my abilities in that skill? Obviously if I had no/all land I'd probably want to mulligan as the opportunity to actually cast some of my spells are worth the 1 card penalty. But what about when I only have 1 land or 1 spell? Are there any good rules of thumb to help me make this decision? How can knowing my deck make these choices easier? What other elements of my hand should I look at when making this decision?
A solid rule of thumb people throw around is to keep the hand if it gives you a plan of action. Do you have something decent to do in the first few turns even if you don't draw what you need?
Beyond this, a good mulligan decision depends on the particular characteristics of your deck. You have to understand what cards you're likely to draw, what your mana curve is, how well it performs with a smaller starting hand, how important tempo is in the first few turns, how important it is to make all your land drops... etc.
I regularly play a couple of different (casual) decks. One is a green ramp deck, where I am often happy to keep a hand with just two lands or one land and some mana acceleration because I'm very likely to draw more mana and have very few spells that cost more than 4. On the other hand, I might mulligan an otherwise healthy hand with 3 or 4 lands but no acceleration, because getting to 3 mana on turn 2 can give me a large tempo advantage. This is very different from my other blue/red deck which has less acceleration and a steeper curve: making at least the first three land drops is critical, so I'd generally mulligan a hand with less than three lands, but worry less about tempo in the first few turns.
Apart from your own deck, you should consider everything you know about your opponent's deck. If you're facing an aggro deck with lots of burn and cheap, fast creatures, an early blocker they can't deal with could be worth more than any other spell in your deck—and you may not have many turns to draw it if it's not in your hand. But against a control deck, that same efficient blocker could be almost worthless, effectively a dead card in your hand.
Dead cards are something you should definitely pay attention to: inevitably, against certain other decks, some of the cards you're playing will be worth a lot less than others. But if you have six good cards and one dead card, mulliganing won't gain you anything, since you'd be going down to six cards anyhow! I'm often tempted to mulligan if I draw two of the same legend or planeswalker, but that's often not the best choice. (And, of course, redundant cards like this can still be useful to mitigate disruption like targeted discard, counterspells and removal.)
Apart from all this general theoretical analysis, you should also just play your deck a whole bunch of times to develop an intuition for which hands are good when. This will also help you identify weird edge case like hands which are perfect... as long as the opponent doesn't have a turn-one lightning bolt or thoughtseize. (My green deck is particularly vulnerable to this when I keep hands with a single land and a mana dork.)
Unfortunately, knowing your deck inside out and comparing your draw against it are the only universal rules that can apply, so it's a highly individual decision that depends on those two factors. If you already have a good idea of the opponent's deck, that also helps, of course. Essentially, experience is where it's at.
You can ask yourself a few questions.
Note that if you often get into a situation where you must consider mulligan (assuming careful shuffling before each game), then there might be an issue with your deck. Adjust the number of mana sources and/or the mana curve of your spells.
On shuffling: obviously, good shuffling is essential to let your deck design have effect on reducing the number of necessary mulligans. During the course of a game, card types tend to clump together as you arrange them: Lands, creatures, and artifacts you play get close to each other, and after the game, they will probably stick together. You have to separate them thoroughly, for example by multiple pile shuffles:
Especially in tournaments, take your time for shuffling, though of course not so much that it could warrant a warning.
Edit: Thanks to thesunneversets for the reminder about first to draw or first to play, +1 for that. Another thing I would like to add on that matter:
If you play it very straight, then the player who is first to play is also the first to announce whether to keep or whether to take a mulligan. Only after the first player has finished all his mulligans and announced to keep his hand, only then does the second player have to start announcing mulligan or not. Therefore, if you are first to draw, that can be a very important factor in your decision whether to mulligan or not. If your opponent has to mulligan, then you can either risk playing a sub-optimal hand because of your card advantage, or you can mulligan as often as he does for a chance of a better quality hand.
On the other hand, if you are first to play and consider mulligan, then you can in a bit of trouble, because everything you say or how long it takes you to say it can be a signal to the opponent. If you take a long time to decide whether to mulligan or not, then this can be understood as a signal that you are not happy with your hand, even if you decide to keep, and that can influence your opponent's choice of mulligan or not.
That way, magic can become truly a mind game like Poker beyond the logic of rules and numbers. This applies to the whole game, not just the mulligan phase.
In the old days, mulliganing was easy. No lands? All lands? Draw a new hand.
And then along came the Paris Mulligan - now a mulligan is always an option, not just a get-out-clause for an absolutely unplayable hand. And while you will certainly still always mulligan an all-lander or a no-lander (unless your deck is EXTREMELY unusual), there are many more borderline cases where you will have to make some tricky choices.
Hackworth has already provided a useful list of things you should be thinking about while deciding whether to take a mulligan, and I've upvoted that, but here's a couple of things that I feel compelled to add:
Beginning players, and I remember well that I used to be like this, have the mentality of "this will be a great hand... if the first card I draw is a land. What the hell, I'll keep it!" Thinking like that probably gives you a 60% chance of losing even before you play a single card, not good odds! Being a card down is bad news, but it's much less bad than running the risk of not being able to play your deck at all. Think through all the probabilities, and then, if necessary, mulligan away. I think it's fair to say that having a good sense when to mulligan is one of THE major factors that distinguishes the pro players in the game...