First of all to see how many revisions you are behind locally, you should do a git fetch to make sure you have the latest info from your remote.
The default output of git status tells you how many revisions you are ahead or behind, but usually I find this too verbose:
$ git status
# On branch master
# Your branch and 'origin/master' have diverged,
# and have 2 and 1 different commit each, respectively.
#
nothing to commit (working directory clean)
I prefer git status -sb:
$ git status -sb
## master...origin/master [ahead 2, behind 1]
In fact I alias this to simply git s, and this is the main command I use for checking status.
To see the diff in the "ahead revisions" of master, I can exclude the "behind revisions" from origin/master:
git diff master..origin/master^
To see the diff in the "behind revisions" of origin/master, I can exclude the "ahead revisions" from master:
git diff origin/master..master^^
If there are 5 revisions ahead or behind it might be easier to write like this:
git diff master..origin/master~5
git diff origin/master..master~5
UPDATE
To see the ahead/behind revisions, the branch must be configured to track another branch. For me this is the default behavior when I clone a remote repository, and after I push a branch with git push -u remotename branchname. My version is 1.8.4.3, but it's been working like this as long as I remember.
As of version 1.8, you can set the tracking branch like this:
git branch --track test-branch
As of version 1.7, the syntax was different:
git branch --set-upstream test-branch