We have used Kindercare, which is a company that only comes to Disney resorts for in-house babysitting. There is also a service called Fairy Godmothers, who will go off-site as well. And lastly, if your kids are 4 and older (and potty-trained) you have the option of using the Disney Kids Clubs, if you are staying at a Disney resort.
We've used both Kindercare and the Kids Clubs. My oldest loved the Sandcastle Club and The Neverland Club (Beach Club and Polynesian, respectively). The Neverland Club is a little more interesting, with more stuff to do, but both are excellent. My child still talks about both of these clubs to this day.
Kindercare is also good, particularly if you have no other option if your kids are under 4, or you want to stay out past 12, when the Kids Clubs close. We used them once, our little one was too young to go to the Kids Club. They are licensed and bonded, and the staff is well qualified. We were given a woman by the name of June, an older woman who was kind of a grandmotherly type. My kids loved her, she brought toys and activities for them, and when we came home they were tucked away in their beds, sound asleep. I plan on using them again in May, and although they are not cheap ($14 per hour for 2 kids), they are definately worth it for peace of mind.