What is a tracking branch?

The ProGit book has a very good explanation:

Tracking Branches

Checking out a local branch from a remote branch automatically creates what is called a tracking branch. Tracking branches are local branches that have a direct relationship to a remote branch. If you’re on a tracking branch and type git push, Git automatically knows which server and branch to push to. Also, running git pull while on one of these branches fetches all the remote references and then automatically merges in the corresponding remote branch.

When you clone a repository, it generally automatically creates a master branch that tracks origin/master. That’s why git push and git pull work out of the box with no other arguments. However, you can set up other tracking branches if you wish — ones that don’t track branches on origin and don’t track the master branch. The simple case is the example you just saw, running git checkout -b [branch] [remotename]/[branch]. If you have Git version 1.6.2 or later, you can also use the --track shorthand:

$ git checkout --track origin/serverfix
Branch serverfix set up to track remote branch refs/remotes/origin/serverfix.
Switched to a new branch "serverfix"

To set up a local branch with a different name than the remote branch, you can easily use the first version with a different local branch name:

$ git checkout -b sf origin/serverfix
Branch sf set up to track remote branch refs/remotes/origin/serverfix.
Switched to a new branch "sf"

Now, your local branch sf will automatically push to and pull from origin/serverfix.

BONUS: extra git status info

With a tracking branch, git status will tell you how far behind your tracking branch you are - useful to remind you that you haven't pushed your changes yet! It looks like this:

$ git status
On branch master
Your branch is ahead of 'origin/master' by 1 commit.
  (use "git push" to publish your local commits)

or

$ git status
On branch dev
Your branch and 'origin/dev' have diverged,
and have 3 and 1 different commits each, respectively.
  (use "git pull" to merge the remote branch into yours)

Below are my personal learning notes on GIT tracking branches, hopefully it will be helpful for future visitors:

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Tracking branches and "git fetch":