[git] How can I delete a file from a Git repository?

The answer by Greg Hewgill, that was edited by Johannchopin helped me, as I did not care about removing the file from the history completely. In my case, it was a directory, so the only change I did was using:

git rm -r --cached myDirectoryName

instead of "git rm --cached file1.txt" ..followed by:

git commit -m "deleted myDirectoryName from git"
git push origin branch_name

Thanks Greg Hewgill and Johannchopin!