Every breed has ailments associated with them. Often Toy varieties can be born with additional medical problems if they're born too small, and don't develop properly. I have a Toy Poodle, and his tear ducts never developed, causing him to tear constantly. This is common with a lot of Toy Poodles.

"Teacup" is not a variety. It is a pet store marketing term. Toy Poodles, for instance, are 10" in height (at the shoulder) or less (in the US). The smallest ones are marketed as "teacups". Quite often they are runts. Acquire these dogs with caution. You can easily become attached after 5 minutes, and might face a dog who's not going to make it and/or will cost thousands in vet bills.