How do I delete a Git branch both locally and remotely?

To delete Locally - (Normal),
 

git branch -d my_branch

If your branch in rebasing/merging progress and that was not done properly means, you will get an error Rebase/Merge in progress so in that case, you won't be able to delete your branch.

So either your need to solve rebasing/merging otherwise you can do force Delete by using,

git branch -D my_branch

To delete in Remote:

git push --delete origin my_branch

can do the same using ,

git push origin :my_branch   # easy to remember both will do the same.