Why is 20 the starting life total for Magic the Gathering? How would the game be different if this number was changed?
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.