You probably know that kids under the age of three get into Disney World parks for free, but did you also know that they eat free and don't count against your room total at your Disney resort? That means that a family of five with a child under the age of three can stay at a value resort like Pop Century, which normally only sleeps four people. Since kids under the age of three can eat free at any Disney table service restaurant, buffets, even character dining, are an especially good deal.
If your child is turning three during your visit, they'll remain free thoughout the trip. Their age for that trip is based on how old they were when they started the trip.
One final thing: I've never heard of anyone being asked by Disney to verify their toddler's age, but if you're worried, bring a copy of their birth certificate.