In simplest terms:
A Dynasty league is one where most (or sometimes all) of your players are kept from year to year with the only additions being from trades, waiver-wire and the yearly rookie draft. Dynasty leagues try to mimic real NFL teams in the way that if you end up with a crappy player, he will either take up a roster spot or you will have to cut him and hope for the best. Good dynasty teams tend to stay good for a number of years, and bad dynasty teams tend to take at least 2-3 seasons to be made competitive again.
A Keeper league is more similar to a regular, run-of-the-mill "redraft" league except that you are able to select "keepers" from your previous year's team. Keepers vary from league to league, but are typically 1-4 players in offense-only leagues, and anywhere up to 10-12 for leagues with IDPs. There are variations to keeper leagues where you forfeit draft choices based on your keeper's draft position last year and such, but as a basic rule it is a league where you can allow to keep a certain amount of players from your previous team without risking losing them in free agency. All non-keepers go into the free-agent pool and are able to be selected by anyone during the draft. Typically in keeper leagues there is only one draft with both non-keepers and rookies all available.
They each have their pros and cons, and it is up to the player to decide which he prefers. Or choose both. I would say as a general rule-of-thumb, dynasty leagues require more research and effort. If you screw up in a keeper league you will usually still have a few decent players and you start back over the next year. If you screw up in a dynasty it could be years before you're ready to play for a championship. Hope this helps.
Thanks to Freyaka for the sig!