Tea Cup Chihuahua, Teacups

 There is actually no official breed called Tea cup Chihuahua or Teacups.

The name is some breeders came up with for the very Tiny Chihuahuas, that are genetically bred to be smaller. This was probable to make the stand out and create more of a personal identity for these small creatures, so they would become more valued. As to Chihuahuas, there are officially only two kinds, Long Coat and Short Coat.
The name "teacup chihuahua" has become so popular lately though that many are considering this to be a breed of their own. One help in their fame has of course been the celebrity Paris Hilton and her famous Chihuahua Tinkabell.
These dogs are really small in size and some even discourage breeders from breeding these kinds of Chihuahuas, at least the really small ones because they can easily become genetically dysfunctional and create medical problems both for themselves and their owners.
The  UK Kennel Club describes a Chihuahua as a compact, alert and graceful swift-moving dog that has terrier-like qualities of temperament. So although Teacup Chihuahua can be small in size, their temper can be something else.
Most people that buy these dogs though tend to buy them because they are easy to handle. Anytime you buy a dog, it is a dog thing to dig into some study of the breed before making a purchase, that will make easier both for you, and the dog in the years to come. People who buy dogs only with their eyes, often tend to end up surprised or disappointed when their pet turned out to have different characteristics than what they had expected.
There are smaller Chihuahuas and larger Chihuahuas, but there are no "tea cup" Chihuahuas. Within a litter even, the sizes of individuals when adult can vary considerably from the very small (micro tiny, runt) to the very large (medium standard, large standard). Teacup chihuahuas can live a very long time if properly cared for from birth. Obviously, getting a good, health tea cup Chihuahua (or any animal in general) depends on finding a good breeder that cares for their animals. Having purchased a tea cup Chihuahua it's important to find a good vet, so that most of these problems can be avoided.
If you really want a very small Chihuahua, you are best to leave buying one until it is at least 4 months old and then even 6 months is better.
At this time it should be clear as to their size and health status.