Executive Summary
$ git push -d <remote_name> <branchname>
$ git branch -d <branchname>
Delete Local Branch
$ git branch -d <branch_name>
$ git branch -D <branch_name>
Delete Remote Branch
As of Git v1.7.0, you can delete a remote branch using
$ git push <remote_name> --delete <branch_name>
which might be easier to remember than
$ git push <remote_name> :<branch_name>