From the comprehensive rules, Rule 104:

104.2. There are several ways to win the game.

104.2a A player still in the game wins the game if all of that player’s opponents have left the game.

104.2b An effect may state that a player wins the game. (In certain multiplayer games, this may not cause the game to end; see rule 104.3h.)

104.2c In a multiplayer game between teams, a team with at least one player still in the game wins the game if all other teams have left the game. Each player on the winning team wins the game, even if one or more of those players had previously lost that game.

104.3. There are several ways to lose the game.

104.3a A player can concede the game at any time. A player who concedes leaves the game immediately. He or she loses the game.

104.3b If a player’s life total is 0 or less, he or she loses the game the next time a player would receive priority. (This is a state-based action. See rule 704.)

104.3c If a player is required to draw more cards than are left in his or her library, he or she draws the remaining cards, and then loses the game the next time a player would receive priority. (This is a state-based action. See rule 704.)

104.3d If a player has ten or more poison counters, he or she loses the game the next time a player would receive priority. (This is a state-based action. See rule 704.)

104.3e An effect may state that a player loses the game.

104.3f If a player would both win and lose simultaneously, he or she loses.

104.3g In a multiplayer game between teams, a team loses the game if all players on that team have lost.

104.3h In a multiplayer game using the limited range of influence option, an effect that states that a player wins the game instead causes all of that player’s opponents within his or her range of influence to lose the game.

104.3i In a tournament game, a player can be awarded a Game Loss or a Match Loss by a judge. See rule 100.6.

These are basically all variations on the same three concepts -- 0 life, no card to draw, or specific text -- but some points are worth mentioning.

First, poison counters - I don't know if there are any Standard cards which give players poison counters, but once that was a viable option for making someone lose.

Secondly, the "0 life" and "no card to draw" conditions both make you lose "when a player gets priority". This means you can pay your last life to make your opponent lose and force a draw -- but only if you pay life as part of a cost, not an effect (since if it were an effect, you'd lose as soon as it resolved since someone would get priority immediately after). The drawing part is also interesting -- if you have to draw more than you have in your library, you draw whatever you got and then lose when a player gets priority. This means you can force a draw with an effect that makes you draw a card when you can't and simultaneously kills your opponent (ideally by making them draw their last card as well, for extra "OMG" credit).

But eventually, in 99% of games these won't matter. Just get to one of the basic three conditions and you're set.

Game loss is also an option in tournaments. But the cases of that happening while playing aren't so common: you'd have to be a bad player haha.

In Magic Online if your clock counts to zero you lose, even if you won the first game of the match and this is the second.

Also if you don't respond for 10 minutes the computer drops you from the tournament.