$ git push -d <remote_name> <branch_name>
$ git branch -d <branch_name>
Note that in most cases the remote name is origin. In such a case you'll have to use the command like so.
$ git push -d origin <branch_name>
Delete Local Branch
To delete the local branch use one of the following:
$ git branch -d branch_name
$ git branch -D branch_name
Note: The -d option is an alias for --delete, which only deletes the branch if it has already
been fully merged in its upstream branch. You could also use -D, which is an alias for --delete --force, which deletes the branch "irrespective of its merged status." [Source: man git-branch]
Also note that git branch -d branch_name will fail if you are currently in the branch you want to remove. The message starts with error: Cannot delete the branch 'branch_name'. If so, first switch to some other branch, for example: git checkout master.
Delete Remote Branch [Updated on 8-Sep-2017]
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>
which was added in Git v1.5.0 "to delete a remote branch or a tag."
Starting on Git v2.8.0 you can also use git push with the -d option as an alias for --delete.
Therefore, the version of Git you have installed will dictate whether you need to use the easier or harder syntax.