Mod stands for modification and basically it is anything that changes any aspect of the game. It may add new functions, fix bugs, add new models, change skins etc. Mods are developed by independent (usually amateur) programmers and designers that are not related to the game developer.
Most modded games
There are mods for nearly each game in series, but THPS3 and THUG2 for the PC version stands out of the row, providing tons of small modifications and several global projects which involves addition of new moves, maps or functions.
This mod changes original game textures to make it look different. This mod type relies mostly to THPS3, there is a lot of skinning mods for this game. Here goes HUD and board skins as well.
There is a variety of mods that changes something in game scipts. It may remove some limits, add new stuff, change game frontend, etc.
This acts as a wrapper for the game executable file allowing user to change some options that he's not allowed to change in default launcher. For the instance such launcher may allow to change game resolution. Depending on the way it was written it may fit the type listed below. In case of THPSHD there was no default launcher at all.
This applies to the game executable file. Most popular is the OpenSpy online fix for every game from THPS4 to THAW on PC that allows to play online. Requires user to replace the original executable.
This does not change anything in the game, but helps user to change it. It may be game data reader/converter, script editor, etc.
To be honest, rare one, there is one for THPS3 called Realistic Sound FX. Soundtrack replacement goes here as well.
This type of mod may include several types listed above to provide better compatibility and stability or to create certain atmosphere by changing several aspects of the game at the same time. LevelMod and THUG PRO fall in this section.
There are 2 main sites about THPS games modding: