How can you legally convert a copyright protected DVD purchased or
rented into an MPEG or AVI file (movie)? I want to edit scenes
together with homemade movies. There are hundreds of movies on file
sharing programs in MPEG format so there must be a program people are
using to convert their DVDs to MPEGs?

Yes there are programs but you cannot legally copy rented movies. You
must own the movie for it to be copied. So by that you are asking a
illegal question on here.

Yes, you've asked an illegal question. Someone call 911.

If the DVD is encrypted (which it will be if it's a major release)
than it is not legal to remove the protection for whatever reason,
even if it would fall under fair use. This is because of the DMCA. The
authors of the law (the contant industry) did what you would call an
end-run around the fair use doctrine. The law says that you can't
bypass a techniclogical measure that prevents unauthorized use of a
copyrighted work. This means that you can copy the encrypted files to
your hard drive but you can't decrypt them (as would be neccary for
you to use them in your project) because that would be an unauthorized
use. The owner of the copyright gets to decide what is unauthorized
use, this means that they can deny you your rights under "fair use"
and possibly "first sale". I suggest reading up on the DMCA at the EFF
http://www.eff.org/IP/DMCA/ for more information on this misguided
law. It is even possible that telling you how to do this (free speech)
may violate the DMCA.

That said, I will answer your question because I believe in fair use.
It is actually quite simple. The video files on a DVD are allready
MPEGs. All you need to do is copy them to the HD and decrypt them.
This is done in one step with a program called a ripper. Go here
http://www.doom9.org/index.html?/dvd_ripper_comparison.htm
and select the one that suits your needs. You can also find more help
on the doom9 site and at
http://www.videohelp.com/

As long as you are using short sceens and don't intende to sell the
resulting product, I would say (altho I am not a lawyer) that you are
well within you fair use rights. However, it would clearly be aginst
plain old copyright law to copy and then return a rented movie.