From what I know, you can't add animations to Skyrim NPCs or your character in third person, you can only replace existing ones.

What about animals and other non-human characters? Could I add a sit animation to a giant, or another attack animation to a mammoth? If not, how are those mods which add entirely new animals possible, like the recent Tropical Skyrim mod which adds raptors?

When a character performs an action such as move forward or jump a specific animation is played. When it is said new animations can't be added what is meant is that new actions can't be added. It is possible to script new actions but the animations for those new actions must be scripted as well.

Ok, thank you, so it looks like adding new animations to animals would be possible with proper scripting. Any way you could point me in the right direction of where to learn more about adding animations?

To do that you will probably need to use http://skyrim.nexusmods.com/mods/11811.