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.

Moving to an existing branch

If you want to move your commits to an existing branch, it will look like this:

git checkout existingbranch
git merge master
git checkout master
git reset --hard HEAD~3 # Go back 3 commits. You *will* lose uncommitted work.
git checkout existingbranch

You can store uncommitted edits to your stash before doing this, using git stash. Once complete, you can retrieve the stashed uncommitted edits with git stash pop

Moving to a new branch

WARNING: This method works because you are creating a new branch with the first command: git branch newbranch. If you want to move commits to an existing branch you need to merge your changes into the existing branch before executing git reset --hard HEAD~3 (see Moving to an existing branch above). If you don't merge your changes first, they will be lost.

Unless there are other circumstances involved, this can be easily done by branching and rolling back.

# Note: Any changes not committed will be lost.
git branch newbranch      # Create a new branch, saving the desired commits
git reset --hard HEAD~3   # Move master back by 3 commits (Make sure you know how many commits you need to go back)
git checkout newbranch    # Go to the new branch that still has the desired commits

But do make sure how many commits to go back. Alternatively, you can instead of HEAD~3, simply provide the hash of the commit (or the reference like origin/master) you want to "revert back to" on the master (/current) branch, e.g:

git reset --hard a1b2c3d4

*1 You will only be "losing" commits from the master branch, but don't worry, you'll have those commits in newbranch!

WARNING: With Git version 2.0 and later, if you later git rebase the new branch upon the original (master) branch, you may need an explicit --no-fork-point option during the rebase to avoid losing the carried-over commits. Having branch.autosetuprebase always set makes this more likely. See John Mellor's answer for details.