There are different kinds of mods: many can be used with only a standard copy of Terraria; some are run independently from the game; others can only be used with a special launcher.
These mods expand the game's items, non-playable characters, or features in one way or another. These mods may or may not require the use of another Mod. Read about each mod for instructions on installation. Always remember to backup files that are called to be overwritten.
Buildaria is distinguished from other map editors by the ability to build in-game, instead of through a program. Among other features, it allows copy and paste, unlimited material use, and a "noclip".
TerraMod is a multi-functional mod to Terraria which adds RPG-like stats and permanent skills to player characters, world generator and building, cheats list, new storage option, and special key binding. This mod is designed to auto update.
Terraria Leveled is a dedicated RPG mod. It adds no items to the vanilla game but instead focuses entirely on character level up through fighting mobs and mining. Mobs and traps have been modified.
Rydian's Terraria Trainer is an indirect mod that runs at the same time as Terraria but in a different instance. This mod contains the following features: become invincible and get infinite item use/placement; obtain huge building range to make creating structures easier; fast-forward time to night; obtain infinite wings or rocket boot fuel; have infinite breath, and many more possibilities.
One of the most well known mods, Calamity is an unofficial expansion to Terraria. Adding several new enemies, over 20 new bosses, and crafting recipes, weapons and even an entirely new lore to the bosses. It adds 2 new difficulties called Revengeance and Death Mode. If you think Expert mode is hard, good luck even getting far in either of those two modes, or even the bosses themselves, since majority of the new bosses are Post-Moon Lord.
These are not actually "mods". Instead, they allow easy access to and manipulation of other mods.
Terraria Custom Content Loader
Terraria Custom Content Loader (TCCL) allows players to add lots of texture, music, and minor functionality changes to Terraria. Includes a handy GUI and helps keep content mods from overlapping.
Detox is a client modification for Terraria that extends the base of what the game can do and adds a layer of customization to the client via the use of plugins. Detox does not modify the client executable that is on disk and does not rely on a specific version of the Terraria.exe, which means that Detox is almost guaranteed to work for any version of Terraria.
Known plugins here
Terraria Online thread here
tAPI is a program that allows its users to add their own content through the use of Hooking and JSON to Terraria 1.2 and later 1.2 updates. This mod is discontinued. It will require a legitimate copy of Terraria (purchased through Steam) to be able to be installed.
Terraria Community Forum thread
tModLoader is a program that lets modders add their own content to Terraria 1.3 and later.
Server mods allow special functions with Terraria multiplayer servers.
TShock is a server modification which contains tools which allow users to summon different mobs (including bosses) and items, change the game time, and teleport to other players in the server.
External tools are mods that function separately from the game. They are installed and opened without the need of the Terraria exe and have their own interface.
Terrasavr is a web-based (no downloading needed) inventory editor to mod a character's current profile including colors, equipment, and inventory.
TerraFirma is a mapping tool used to view the Terraria world. It shows where all ores and chests are located, as well as the contents of chests. It can also be zoomed in and out.
tEdit is a modification as well as a mapping program that enables the user to create chests, blocks, walls, and practically anything that comes to mind. It is also popular because of its ability to create worlds without even starting Terraria (similar to World Viewer). Similar to TerraFirma, it can be zoomed in/out and see where ore and chests are located. Unlike Terrafirma, tEdit allows ore to be created/deleted and chest contents to be modified.
Dean's Custom World Generator
Dean's Custom World Generator will allow a user to slightly alter the world generator's algorithms to make a still randomized world with or without the features the user desires. This tool is still in development.
Terraria Save Game Editor
Terraria Save Game Editor (TSGE) is a program to mod Terraria characters. Characters can have all their stats changed (i.e. changed from medium core to softcore), manage and edit the inventory, and edits the character buffs/debuffs.
These mods alter user interface.
The Inventory Manager mod aids players in sorting and crafting new items with lots of hotkeys and sorting buttons.
XBox Remote on PC
These mods change the textures of the game or UI without any gameplay changes. For more textures, please visit the Terraria Online Forums.
The Terraria-Interface-Mod by Testing_1_2 sharpens up the inventory and font in Terraria to make a very easy to see UI.
Svard's Fonts and Cursor
Svard's Fonts and Cursor will allow the player to choose from an assortment of fonts and cursors to fit his or her liking.
hq2x is a (minor) complete texture overhaul.
Solstice's HD texture pack
Solstice's HD texture pack provides a more granular, high contrast texture for the entire game. (Pre-1.2, but the author is working on it.)