How do I rename all files to lowercase?

If you're comfortable with the terminal:

1. Open Terminal.app, type cd and then drag and drop the Folder containing the files to be renamed into the window.
2. To confirm you're in the correct directory, type ls and hit enter.

3. Paste this code and hit enter:

   for f in *; do mv "$f" "$f.tmp"; mv "$f.tmp" "`echo $f | tr "[:upper:]" "[:lower:]"`"; done
   

4. To confirm that all your files are lowercased, type ls and hit enter again.

(Thanks to @bavarious on twitter for a few fixes, and thanks to John Whitley below for making this safer on case-insensitive filesystems.)

The question as-asked is general, and also important, so I wish to provide a more general answer:

Simplest case (safe most of the time, and on Mac OS X, but read on):

for i in * ; do j=$(tr '[:upper:]' '[:lower:]' <<< "$i") ; mv "$i" "$j" ; done

You need to also handle spaces in filenames (any OS):

IFS=$'\n' ; for i in * ; do j=$(tr '[:upper:]' '[:lower:]' <<< "$i") ; mv "$i" "$j" ; done

You need to safely handle filenames that differ only by case in a case-sensitive filesystem and not overwrite the target (e.g. Linux):

for i in * ; do j=$(tr '[:upper:]' '[:lower:]' <<< "$i") ; [ -e "$j" ] && continue ; mv "$i" "$j" ; done 

Note about Mac OS X:

Mac's filesystem is case-insensitive, case-preserving.

There is, however, no need to create temporary files, as suggested in the accepted answer and comments, because two filenames that differ only by case cannot exist in the first place, ref.

To show this:

$ mkdir test
$ cd test
$ touch X x
$ ls -l 
total 0
-rw-r--r--  1 alexharvey  wheel  0 26 Sep 20:20 X
$ mv X x
$ ls -l 
total 0
-rw-r--r--  1 alexharvey  wheel  0 26 Sep 20:20 x