An updated answer reflecting changed times:
GitPython currently is the easiest to use. It supports wrapping of many git plumbing commands and has pluggable object database (dulwich being one of them), and if a command isn't implemented, provides an easy api for shelling out to the command line. For example:
repo = Repo('.')
repo.checkout(b='new_branch')
This calls:
bash$ git checkout -b new_branch
Dulwich is also good but much lower level. It's somewhat of a pain to use because it requires operating on git objects at the plumbing level and doesn't have nice porcelain that you'd normally want to do. However, if you plan on modifying any parts of git, or use git-receive-pack and git-upload-pack, you need to use dulwich.