NOTE: This answer changes SHA1s, so take care when using it on a branch that has already been pushed. If you only want to fix the spelling of a name or update an old email, git lets you do this without rewriting history using.mailmap. See my other answer.

Using Interactive Rebase

You could do

git rebase -i -p <some HEAD before all of your bad commits>

Then mark all of your bad commits as "edit" in the rebase file. If you also want to change your first commit, you have to manually add it as the first line in the rebase file (follow the format of the other lines). Then, when git asks you to amend each commit, do

git commit --amend --author "New Author Name <email@address.com>"

edit or just close the editor that opens, and then do

git rebase --continue

to continue the rebase.

You could skip opening the editor altogether here by appending --no-edit so that the command will be:

git commit --amend --author "New Author Name <email@address.com>" --no-edit && \
git rebase --continue

Single Commit

As some of the commenters have noted, if you just want to change the most recent commit, the rebase command is not necessary. Just do

git commit --amend --author "New Author Name <email@address.com>"

This will change the author to the name specified, but the committer will be set to your configured user in git config user.name and git config user.email. If you want to set the committer to something you specify, this will set both the author and the committer:

git -c user.name="New Author Name" -c user.email=email@address.com commit --amend --reset-author

Note on Merge Commits

There was a slight flaw in my original response. If there are any merge commits between the current HEAD and your <some HEAD before all your bad commits>, then git rebase will flatten them (and by the way, if you use GitHub pull requests, there are going to be a ton of merge commits in your history). This can very often lead to a very different history (as duplicate changes may be "rebased out"), and in the worst case, it can lead to git rebase asking you to resolve difficult merge conflicts (which were likely already resolved in the merge commits). The solution is to use the -p flag to git rebase, which will preserve the merge structure of your history. The manpage for git rebase warns that using -p and -i can lead to issues, but in the BUGS section it says "Editing commits and rewording their commit messages should work fine."

I've added -p to the above command. For the case where you're just changing the most recent commit, this is not an issue.

Update for modern git clients (July 2020)

Use --rebase-merges instead of -p (-p is deprecated and has serious issues).