Change your files

This step should be blatantly obvious, but some users still skip it. If you committed a password, change it! If you committed a key, generate a new one. If you commited private files remove them.
Once the commit has been pushed you should consider the data to be compromised.

Purge the file from your repo

Now that the password is changed, you want to remove the file from history and add it to the.gitignore to ensure it is not accidentally re-committed. For our examples, we're going to removeRakefile from the GitHub gem repo.
$ git clone https://github.com/defunkt/github-gem.git
# Initialized empty Git repository in /Users/tekkub/tmp/github-gem/.git/
# remote: Counting objects: 1301, done.
# remote: Compressing objects: 100% (769/769), done.
# remote: Total 1301 (delta 724), reused 910 (delta 522)
# Receiving objects: 100% (1301/1301), 164.39 KiB, done.
# Resolving deltas: 100% (724/724), done.

$ cd github-gem

$ git filter-branch --index-filter 'git rm --cached --ignore-unmatch Rakefile' \
  --prune-empty --tag-name-filter cat -- --all
# Rewrite 48dc599c80e20527ed902928085e7861e6b3cbe6 (266/266)
# Ref 'refs/heads/master' was rewritten
This command will run the entire history of every branch and tag, changing any commit that involved the file Rakefile, and any commits afterwards. Commits that are empty afterwards (because they only changed the Rakefile) are removed entirely. Now that we've erased the file from history, let's ensure that we don't accidentally commit it again.
Please note that this will overwrite your existing tags.
$ echo "Rakefile" >> .gitignore

$ git add .gitignore

$ git commit -m "Add Rakefile to .gitignore"
# [master 051452f] Add Rakefile to .gitignore
#  1 files changed, 1 insertions(+), 0 deletions(-)
This would be a good time to double-check that you've removed everything that you wanted to from the history. If you're happy with the state of the repo, you need to force-push the changes to overwrite the remote repo.
$ git push origin master --force
# Counting objects: 1074, done.
# Delta compression using 2 threads.
# Compressing objects: 100% (677/677), done.
# Writing objects: 100% (1058/1058), 148.85 KiB, done.
# Total 1058 (delta 590), reused 602 (delta 378)
# To https://github.com/defunkt/github-gem.git
#  + 48dc599...051452f master -> master (forced update)
You will need to run this for every branch and tag that was changed. The --all and --tags flags may help make that easier.

Cleanup and reclaiming space

While git filter-branch rewrites the history for you, the objects will remain in your local repo until they've been dereferenced and garbage collected. If you are working in your main repo you might want to force these objects to be purged.
$ rm -rf .git/refs/original/

$ git reflog expire --expire=now --all

$ git gc --prune=now
# Counting objects: 2437, done.
# Delta compression using up to 4 threads.
# Compressing objects: 100% (1378/1378), done.
# Writing objects: 100% (2437/2437), done.
# Total 2437 (delta 1461), reused 1802 (delta 1048)

$ git gc --aggressive --prune=now
# Counting objects: 2437, done.
# Delta compression using up to 4 threads.
# Compressing objects: 100% (2426/2426), done.
# Writing objects: 100% (2437/2437), done.
# Total 2437 (delta 1483), reused 0 (delta 0)
Note that pushing the branch to a new or empty GitHub repo and then making a fresh clone from GitHub will have the same effect.

Dealing with collaborators

You may have collaborators that pulled your tainted branch and created their own branches off of it. After they fetch your new branch, they will need to use git rebase on their own branches to rebase them on top of the new one. The collab should also ensure that their branch doesn't reintroduce the file, as this will override the .gitignore file. Make sure your collab uses rebase and not merge, otherwise he will just reintroduce the file and the entire tainted history... and likely encounter some merge conflicts.

Cached data on GitHub

Be warned that force-pushing does not erase commits on the remote repo, it simply introduces new ones and moves the branch pointer to point to them. If you are worried about users accessing the bad commits directly via SHA1, you will have to delete the repo and recreate it. If the commits were viewed online the pages may also be cached. Check for cached pages after you recreate the repo, if you find any open a ticket on GitHub Support and provide links so staff can purge them from the cache.

Avoiding accidental commits in the future

There are a few simple tricks to avoid committing things you don't want committed. The first, and simplest, is to use a visual program like GitHub for Mac or gitx to make your commits. This lets you see exactly what you're committing, and ensure that only the files you want are added to the repo. If you're working from the command line, avoid the catch-all commands git add . and git commit -a, instead use git add filename and git rm filename to individually stage files. You can also use git add --interactive to review each changed file and stage it, or part of it, for commit. If you're working from the command line, you can also use git diff --cached to see what changes you have staged for commit. This is the exact diff that your commit will have as long as you commit without the -a flag.

Other reading