There are just three Poodle varieties, and they are all based on size. Those regarding the coat style are not recognized as variations by the big kennel clubs such as the AKC.
Coat type is a type of Poodle that is not really recognized as a variety. There are the curly coats, and there are the corded coats. (Wavy is not a proper description for this breed, it should be noted.) The curly version is the one we've all seen. The corded Poodle coat is where the curls are like certain sheepdog breeds. With the corded Poodle, the hair grows into rope-like strands, and these are called "dreds."
Due to this breed's huge popularity explosion, based primarily in the dogs' hypoallergenic coat, there are a great many sizes. Many terms are used to describe the sizes, and most of these terms are not standardized or recognized, universally or otherwise. The formally known terms are the Standard Poodle, which is the largest of the three variations. The middle size is the Miniature Poodle and is medium in its height. The smallest of the three is the Toy. It is the Toy Poodle that has nearly all the non-standard names. This varietal is the most popular and may be called the teacup (or Tcup), mini, micro, petite, dwarf, pocket Poodle, etc.
There is a growing crowd that is hoping to introduce a new variation, the king Poodle. These dogs are giant in size and are significantly larger than the Standard. For the most part , the king is not presently recognized other than by people seeking to breed and sell them.