Terraria is an action adventure sandbox game with many recognizable mechanics like crafting, building, and combat. At the start of the game you customize your character with the many options available and then generate a world. From there it is up to you to explore, loot, and make your character stronger to prepare for boss fights. Defeating bosses is how you progress through Terraria. This site contains information about the bosses that must be defeated to unlock Hardmode, which marks the beginning of the game's more challenging second phase.
The Eye of Cthulhu is the first boss you need to defeat to progress. This fight can take place in any biome in your world, but only at night time. Once a player has obtained over 200 HP, the Eye of Cthulhu will have a 1/3 chance of spawning at the start of each night. This boss doesn't necesarily drop anything too important. Once defeated you will be awarded some demonite/crimtane ore, some healing potions, money, and maybe a couple of other items that aren't that common.
The next boss in the progression will depend on what evil biome you have in your world. If you generated your world with the Crimson biome, you will have to fight the Brain of Cthulhu. If you generate your world with the Curruption biome, you will have to fight the Eater of Worlds. To spawn these bosses, you must find and break 3 orbs in their respective biomes. You must fight these bosses in their specific biomes. The main drops from these bosses are crimtane/demonite ore as well as the required materials to turn that ore into armor and tools.
For the next boss on the list you must travel near one of the edges of your world and find the dungeon. Here, there will be an old man, if you speak to this old man at night it will summon Skeletron. The main incentive of killing this boss isn't its drops, it is the access to the dungeon you gain after. Before Skeletron is killed if a player attempts to enter the dungeon they will be killed by a dungeon guardian immediately. Once in the dungeon you'll find chests full of weapons and accesories to further power up your character.
The final boss that needs to be defeated before entering Hard mode is the Wall of Flesh. This monstrosity marks the end of the first half of the game. Once defeated stronger enemies, new biomes, and harder bosses will be unleashed onto your world. To summon the Wall of Flesh you must find a voodoo doll in the underworld and throw it into the lava, sacrificing one of your freindly NPC's to commence the epic battle. This boss drops a variety of weapons and accesories that will help you in the early stages of Hard mode.
The Eye of Cthulhu
The Brain of Cthulhu / The Eater of Worlds
Skeletron
The Wall of Flesh
One of the things that sets Terraria apart from other similar games are its epic boss battles. While there are many other Pre-Hard mode bosses, these are the only ones required for the games progression. You can learn more about the optional bosses, Hard mode, and more about Terraria in general with the links at the top of the page.