How to remove a directory from git repository?

I have 2 directories on my GitHub repository. I'd like to delete one of them. How could I do that without deleting and re-creating entire repository?

Remove directory from git and local

You could checkout 'master' with both directories;

git rm -r one-of-the-directories // This deletes from filesystem
git commit . -m "Remove duplicated directory"
git push origin <your-git-branch> (typically 'master', but not always)

Remove directory from git but NOT local

As mentioned in the comments, what you usually want to do is remove this directory from git but not delete it entirely from the filesystem (local)

In that case use:

git rm -r --cached myFolder

To remove folder/directory only from git repository and not from the local try 3 simple commands.

Steps to remove directory

git rm -r --cached FolderName
git commit -m "Removed folder from repository"
git push origin master

Steps to ignore that folder in next commits

To ignore that folder from next commits make one file in root folder (main project directory where the git is initialized) named .gitignore and put that folder name into it. You can ignore as many files/folders as you want

.gitignore file will look like this

/FolderName

If, for some reason, what karmakaze said doesn't work, you could try deleting the directory you want using or with your file system browser (ex. In Windows File Explorer). After deleting the directory, issuing the command:
git add -A
and then
git commit -m 'deleting directory'
and then
git push origin master.