Photo backups are like backing up any other data, and so the same principles from computing apply:

• You want to have one active copy. This would be your memory card and/or computer hard drive when you're editing/organizing.
• You want to have one easy-to-access backup. This is so that you can get the safe copy in the event that you have a minor crash or corrupted file. (This includes corruption by overzealous editing). An external hard drive works well; you can also get a home server, NAS, etc. Burnt DVDs or archival memory cards work too.
• You want to have one offsite backup. This is to guard against catastrophic losses (ie: your house burns down) which take your easy-to-access backup with it. It also guards against coincidental failure of your active + first backup copy; this is common since you often don't check your first backup for errors until your active copy fails. Online backup services are great for this, but burnt DVDs or memory cards in a safety deposit box work too.

General rule: the more copies you have, the better.