If you want more detailed explanations of the following commands, then see the long answers in the next section.
Deleting a remote branch:
git push origin --delete <branch> # Git version 1.7.0 or newer
git push origin :<branch> # Git versions older than 1.7.0
Deleting a local branch:
git branch --delete <branch>
git branch -d <branch> # Shorter version
git branch -D <branch> # Force delete un-merged branches
Deleting a local remote-tracking branch:
git branch --delete --remotes <remote>/<branch>
git branch -dr <remote>/<branch> # Shorter
git fetch <remote> --prune # Delete multiple obsolete tracking branches
git fetch <remote> -p # Shorter
More here: https://stackoverflow.com/a/23961231