How to remove a file from the staging area (= index = cache) in Git?

EDIT This question can be understood in two ways, and the optimal answer is different in the two cases.

• Question 1: I added a previously untracked file to the staging area. How can I remove this file from the staging area without removing it from the file system?

  Answer 1: Use the following command, as described in John Feminella’s answer:

  git rm --cached <file>
  

• Question 2: I modified a file already tracked, and added my modifications to the staging area. How can I remove my modifications from the staging area? I.e., how can I unstage my modifications in the file?

  Answer 2: Use the following command, as described in David Underhill’s answer:

  git reset <file>