Many people find the copyright laws to be confusing and often times misinterpret USC Title 17 (Copyright Laws). This particular matter falls under fair use (Chapter 1 § 108) .
Let me simplify it all by saying it is perfectly legal to make a backup copy of a CD or DVD so that you can continue to enjoy the copyrighted material if your original copy fails (i.e. scratches, warps, etc.). It is illegal to make copies of CDs or DVDs if you intend to distribute them to third parties, even by giving them away, or to make copies and then dispose of or sell the original media.
You must maintain the originally purchased content to have authorization to hold a digital backup of that content. Each client must sign a consent to liability form for any conversion job stating they understand this.
For content you produced yourself (e.g. home movies) you have sole discretion over authorized reproductions as well as the responsibility of filing for copyright if desired. If the content within contains copyright material (e.g. recording a band's performance live) you may need to gain permissions for reproduction of that material, which is why youtube removes content regularly.