How do I find which rpm package supplies a file I’m looking for?

This is an old question, but the current answers are incorrect 🙂

Use yum whatprovides, with the absolute path to the file you want (which may be wildcarded). For example:

yum whatprovides '*bin/grep'

Returns

grep-2.5.1-55.el5.x86_64 : The GNU versions of grep pattern matching utilities.
Repo        : base
Matched from:
Filename    : /bin/grep

You may prefer the output and speed of the repoquery tool, available in the yum-utils package.

sudo yum install yum-utils
repoquery --whatprovides '*bin/grep'
grep-0:2.5.1-55.el5.x86_64
grep-0:2.5.1-55.el5.x86_64

repoquery can do other queries such as listing package contents, dependencies, reverse-dependencies, etc.