Three-Headed In-Fighting

Here's the best way I know to play Magic with three players, a format I improvised over a decade ago during a Summer when my draft club was just three people and we were desperate.

Only one player will win, but you have partners: You and the player to your left are partners. You and the player to your right are also partners. Finally, both your opponents are partners. Partners how? On defense… In the exact same way that two-headed giant partners share a life total and block against attacks together.

How do you win? By dropping your opponents' shared life total to 0 before either of your own shared life totals drop to 0. That's right, you have two life totals, and neither are solely your own. If you gain life, both your totals increase. If you pay life, both your totals drop. If you deal damage to an opponent, or cause them to lose life, only the life total they share with your other opponent is affected. If an effect touches all players, it touches each life total once. (This is different than 2HG and you're welcome to play it that way instead, but this way is less disruptive.) If an effect hits all opponents, it hits only the life total you don't share in. If you redirect damage from a player to a planeswalker, both that player's life totals are spared.

Otherwise, the rules are the same as normal Magic.  Life totals start at 20. You take your own turns, and you take them in clockwise order. It's multiplayer, so there's only one game per match, each player gets one free mulligan, and the first player draws a card. Cards that refer to "you" mean only you, not either of your partners. Cards that refer to opponents or teammates apply to all the other players, as they are both. You can't lend cards or mana to other players (unless a card tells you to).

An example:
I'm playing mono-green and currently control Elvish Mystic and Rumbling Baloth. You're playing mono-white and control Soul Warden and Mistral Charger. Lorelei is playing red-green and controls Zo-Zu the Punisher. All our shared life totals are at 20.

I attack with my 4/4, and you and Lorelei confer: Should you take the 4 damage or double-block with Charger and Zo-Zu? I might have a trick, and neither of you wants to lose your creature, so you don't block. Your shared life total goes to 16, and I'm winning.

On your turn, you play a Plains, triggering Zo-Zu, who pings the life total you share with me to 18 and the one you share with Lorelei to 14. You then cast Raise the Alarm, triggering Soul Warden twice, gaining 2 life for both your totals, undoing Zo-Zu's punishment. Your Mistral Charger attacks; neither my elf nor Lorelei's shaman can block it, and so our shared life total drops to 18.

On Lorelei's turn, she plays a forest, triggering Zo-Zu and dropping her totals to 16 (her and me), and 12 (you and her). She plays Hurricane with X=3 to kill your pegasus, and all three life totals drop by 3 (her and me: 13, you and her: 9, you and me: 15). She attacks with Zo-Zu. You could block with both your soldier tokens, but you decide it's unfair for you to pay that burden and say you'll block with one soldier if I block with my elf. I look at my hand of all land, despair, and agree. Both our 1/1s trade with her 2/2.

You continue to gain life (for you and me, and for you and Lorelei) over the next few turns by churning out token creatures. Your army swarms past my Baloth, knocking the life total Lorelei and I share down to 0. You win the game.

Because a player has to overcome two players' defenses to push damage through, Three-Headed In-Fighting takes longer than a duel, but it's a multiplayer format that greatly mitigates politics and helps ensure a good game (because even if you stumble, each of your opponents will pick up the slack for you to keep themselves alive) with no player elimination.

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Other Three Player Formats:

Archenemy is another excellent option. One player takes on all the other players, which sounds pretty one-sided but they also get a deck of schemes that balance the scales very well. Archenemy was released long after I devised Three-Headed In-Fighting, and is a superior three-player format, if you're interested in assymetic play and can get your hands on the product.

Free-for-All is the oldest multiplayer format, hailing all the way back to Alpha, even before duels became the default way to play Magic. At 4+ players, it works just fine provided you love political manuevering, and don't mind waiting a while for your turn. It technically works for three-players, but three-player politics are often brutal and commonly lead to king-making; I can't recommend this format for three players at all.

Attack-Left is another very old multiplayer format, and it handles politics pretty well:  You can only attack the player on your left (and so only block attacks from the player on your right). Your spells can't directly affect players further than those two opponents, but such players don't exist in three-player. The downside here is that one player will be eliminated before the game ends, and it can easily be a long time before what-has-effectively-become-a-duel does.

I would be remiss not to mention Commander, Planechase, Conspiracy, and Take the Crown, all of which can layer onto any of the four formats I've already detailed, and add a lot of spice. Both Conspiracy sets also help address the multiplayer-only problems of king-making and turtling.