I always have to look this up.. here is hoping that typing it out will help me remember.
To delete the remote branch
git push --delete <remote-name> <branch-name>
To delete the local branch is just as easy
git branch -d branch-name
Git is going to sanity check you with a lowercase flag because deleting a branch is a big deal, so if you're sure just go ahead and use the flag with a capital letter.
git branch -D branch-name