One of the great things about Terraria is that the order of bosses that you defeat is pretty flexible. Some bosses are even completely optional as far as progression goes. That being said, most players will usually beat bosses in the following order:

Main Bosses

Pre-Hardmode

1. Eye of Cthulhu. Either this boss spawns on its own after a while, or you will spawn it yourself using a Suspicious-Looking Eye. The Eye has the least health of any boss, and has only three distinct attach patterns.

2. Eater of Worlds/Brain of Cthulhu (depending on Corrupted or Crimson world). Summon this boss by destroying three Shadow Orbs/Crimson Hearts.

3. Skeletron. He blocks the entrance to the dungeon. If you don't kill Skeletron, a Dungeon Guardian will spawn anytime you try to enter the dungeon. DG will one-shot you even if you have the best armor in the game.

4. Wall of Flesh. After throwing a voodoo guide doll into lava in the underworld, this boss will spawn. Beating WoF activates hardmode and allows you to summon new bosses.

[These bosses can be beaten in any order, but this is the order I've always done them in. Also, technically Eye of Cthulhu isn't required, but he shares the same ore drop as Eater of Worlds/Brain of Cthulhu, and the ore is required to make one of the better swords in the game.]

Post-Hardmode

5. The Destroyer. A huge worm type boss, most people face him before any other mechanical boss because of easy access to several piercing weapons that make him really easy to beat (like Nimbus Rod or Daedalus Bow).

6. The Twins. Somewhat more difficult. Usually requires an aerial arena to beat.

7. Skeletron Prime. Pretty tough, since he has 4-5 attacks that he performs simultaneously. You better be good at dodging if you want to beat him.

[Note that the mechanical bosses can be beaten in any order.]

8. Plantera. Once all three mechanical bosses have been beaten at least once, Plantera Bulbs will spawn in the jungle. Break one to spawn this boss. Once Plantera is defeated, it drops a temple key (which can be used to open the Jungle temple) and allows more monster types to spawn in the dungeon.

9. Golem. This boss is found in the jungle temple and can be summoned by using a solar cell on the altar at the bottom of the temple. Be sure to disable any traps in the room first, otherwise the fight will get quite tough!

10. Lunatic Cultist. Once the Golem is defeated, several cultists will spawn near the dungeon, praying to some symbol. Kill these cultists to spawn the Lunatic Cultist, who absorbs the symbol's strength and fights you in the air. Beating the Lunatic Cultist will spawn four Lunar Pillars in your world.

11. Lunar Pillars. Not a boss per se, but still a challenging series of fights. Each pillar has its own set of enemies and obstacles. You'll know you're close to one when the colors on your screen start changing to the color of the pillar you're close to. The pillars do not despawn and remain on the map until destroyed.

12. Moon Lord. The hardest boss in the game (so far). Destroying the four Lunar Pillars will cause him to spawn after about 30 seconds to 1 minute (don't remember the exact timing). A definite challenge even with the best armor in the game. Getting help from friends is not required but recommended.

Optional Bosses

• King Slime. Hard to kill right away, but pretty easy once you've acquired some decent beginner gear. Spawns at the end of a slime rain event if you've killed enough slimes.

• Queen Bee. Occasionally in the jungle you'll find bee hives. Inside the hives is a 2x2 block that, once destroyed, spawns the Queen Bee. Drops various bee-related items when defeated.

• Goblin Invasion. An event in which a bunch of goblins will attack at once. Has a random chance of happening in the morning once the player has more than 200 health.

• Snowman Invasion. An event that happens when the player uses a snowglobe (available from presents that only drop during the annual Christmas event - see this question for more details on how to get it year-round). A bunch of snowmen will attack in waves.

• Pirate Invasion. An event where pirates attack in waves. Good source of money and gold furniture drops. Careful of the captain, he packs a punch!

• Queen Slime. The hardmode equivalent to the King Slime fight. She spawns a ton of slimes that shoot many, many projectiles. .

• Duke Fishron. This boss will only spawn when you fish in the ocean while using a truffle worm as bait. He's fast, tough, and drops some really good gear. An arena is highly encouraged.

• Empress of Light. In the Hallow, between 7:30pm to midnight, you'll occasionally see a rainbow butterfly called a Prismatic Lacewing. If you kill it, you'll summon the Empress of Light, a challenging boss with a spectrum of colorful attacks. She drops some of the best gear in the game, but the fight is quite difficult - especially if you're going for the Terraprisma!

• Pumpkin Moon. A Halloween-themed invasion event where the player has to beat as many monsters in one night as possible. Getting to higher waves yields greater rewards, but also greater risk of death. The "end boss" of this event is the Pumpking.

• Frost Moon. A Christmas-themed invasion event where the player has to beat as many monsters in one night as possible. Getting to higher waves yields greater rewards, but also greater risk of death. The "end boss" of this event is the Frost Queen.

• Martian Madness. An invasion event where aliens attack. This event is triggered by letting a martian probe (found in upper atmosphere area) see you and escape.

• Old One's Army. A special Dungeon Defenders 2 crossover event that can be started by placing an Eternia crystal in a crystal stand (both purchased from the tavernkeep NPC). The difficulty of the event varies depending on how far in the game you've progressed. The "final boss" of this event is Betsy, though she only appears if you've beaten the Golem.