Most all bosses are fought by summoning them through crafted items and at specific times during the day. They are listed below in a general order of weakest to hardest. For some bosses in single player, if you die during the fight, you'll have to resummon and start over. Same goes for multiplayer if everyone dies at the same time and the boss has no one in its immediate vicinity to attack. Other bosses will not reset and will most likely meet up with you at your spawn point when you die, allowing you to respawn and resume your attack. Most bosses also have a time limit, requiring you to fight quickly and efficiently before you run out of time.

King Slime Spawns very rarely during daytime in Forests that are not in the center third of the map (so basically they can't spawn around your spawn point). Can also be summoned with a Slime Crown, crafted from a Gold Crown and 99 Gel at a Demon Altar. Can be summoned at anytime, anywhere (space permitted), and with no time limit. Super easy and not really considered a boss by most, but still counts.
Eye of Cthulhu Has a chance of spawning at night once the player has at least 200 health, at least 10 Defense, and has 3 or more NPCs housed, but only once. Can also be summoned anywhere by using a Suspicious Looking Eye at night, crafted from 6 Lens at a Demon Altar. Will despawn if you die, and must be killed before sunrise.
Eater of Worlds Summoned by either smashing 3 Shadow Orbs, or using Worm Food, which is crafted from 30 Vile Powder and 15 Rotten Chunks at a Demon Altar. Can only be summoned and must be fought in the Corruption. Will despawn if you die, but has no time limit. Be careful that you don't move out of the Corruption or it will despawn.
Skeletron Summoned by speaking to the Old Man at the Dungeon entrance at night. Will despawn if you die. If you can't kill it before sunrise, he will insta-kill you. Can only kill Skeletron once per world.
Wall of Flesh The last and toughest of the Normal bosses. Summoned by dropping a Guide Voodoo Doll into a pool of lava in the Underworld. Once you throw your doll into the lava, your Guide will be sacrificed and you will receive the Horrified debuff, which keeps you close to the Wall of Flesh and will pull other players to it when they move too far. Basically, this debuff locks all players into the fight until they or the Wall of Flesh die. The boss is a literal wall that will push you through your entire world until it gets to the end, which will then crush and kill you automatically if you're still alive. When messing around in the Underworld, you must be careful that you don't let a Voodoo Demon drop the Guide Voodoo Doll into lava upon death, or else it will summon the Wall of Flesh.
The Destroyer A Hardmode-only boss that is summoned anywhere at night by using a Mechanical Worm at night, crafted from 7 Souls of Night, 6 Rotten Chunks, 5 Copper Bars, and 5 Iron Bars at a Mythril Anvil. Must be defeated before sunrise or else it will despawn. Does not despawn when you are killed.
The Twins A Hardmode-only boss that is summoned anywhere by using a Mechanical Eye at night, crafted from 3 Lens, 5 Copper Bars, 5 Iron Bars, and 7 Souls of Light. Must be defeated before sunrise or else it will despawn. Does not despawn when you are killed.
Skeletron Prime A Hardmode-only boss that is summoned anywhere by using a Mechanical Skull at night, crafted from 30 Bones, 5 Copper Bars, 5 Iron Bars, 5 Souls of Light, and 5 Souls of Night. Dying will cause it to despawn. If you haven't killed him by sunrise, he will go into "Godmode" and begin to insta-kill you if you stand still. He will also increase his defense, only taking 1 damage from regular hits and 2 from criticals.
Ocram The console-exclusive boss that is summoned by using a Suspicious Looking Skull, crafted from 1 Mechanical Skull, 2 Mechanical Eyes, 20 Adamantite Bars, 10 Souls of Light, and 10 Souls of Night. Can only be summoned at night.
Boss strategy for Wall of Flesh: