You can undo git add before commit with

git reset <file>

which will remove it from the current index (the "about to be committed" list) without changing anything else.

You can use

git reset

without any file name to unstage all due changes. This can come in handy when there are too many files to be listed one by one in a reasonable amount of time.

In old versions of Git, the above commands are equivalent to git reset HEAD <file> and git reset HEAD respectively, and will fail if HEAD is undefined (because you haven't yet made any commits in your repository) or ambiguous (because you created a branch called HEAD, which is a stupid thing that you shouldn't do). This was changed in Git 1.8.2, though, so in modern versions of Git you can use the commands above even prior to making your first commit:

"git reset" (without options or parameters) used to error out when you do not have any commits in your history, but it now gives you an empty index (to match non-existent commit you are not even on).

Documentation: git reset

Undo a commit & redo

$ git commit -m "Something terribly misguided" # (0: Your Accident)
$ git reset HEAD~                              # (1)
[ edit files as necessary ]                    # (2)
$ git add .                                    # (3)
$ git commit -c ORIG_HEAD                      # (4)

1. This command is responsible for the undo. It will undo your last commit while leaving your working tree (the state of your files on disk) untouched. You'll need to add them again before you can commit them again).

2. Make corrections to working tree files.

3. git add anything that you want to include in your new commit.

4. Commit the changes, reusing the old commit message. reset copied the old head to .git/ORIG_HEAD; commit with -c ORIG_HEAD will open an editor, which initially contains the log message from the old commit and allows you to edit it. If you do not need to edit the message, you could use the -C option.

Alternatively, to edit the previous commit (or just its commit message), commit --amend will add changes within the current index to the previous commit.

To remove (not revert) a commit that has been pushed to the server, rewriting history with git push origin master --force is necessary.

Further Reading

How can I move HEAD back to a previous location? (Detached head) & Undo commits

The above answer will show you git reflog, which you can use to determine the SHA-1 for the commit to which you wish to revert. Once you have this value, use the sequence of commands as explained above.

HEAD~ is the same as HEAD~1. The article What is the HEAD in git? is helpful if you want to uncommit multiple commits.