Bubble tea goes by many names: boba tea, momi, pearl tea....
There really is no right or wrong name.
Invented in Taiwan in the 1980's, bubble tea has become much more than the traditional milk, tea, flavoring, and tapioca ball concoction. Bubble tea has since become sort of a phenomenon, spreading throughout Asia and many other countries around the world. While I don't know the full geographic extent of its popularity, bubble tea is now widely found throughout the mainland U.S. Most often found in Chinatowns, Japantowns, Koreatowns, and areas of the same ilk.
I think part of bubble tea's popularity derives from its variability. It can be crafted to any individual's liking and the flavor options can be tailored to local tastes.
Growing up in Honolulu, Hawaii I experienced the introduction and explosion of the bubble tea phenomenon. I am not exactly sure when I had my first bubble tea, but I think it was from Pearl's Korean BBQ in Manoa Marketplace. Unfortunately, I remember their bubble tea wasn't very good. The tapioca balls were a bit hard and they used powder flavorings (not sure how it is now, or if they still make bubble tea there). But that was then. Now in Hawaii bubble tea is as common as shave ice. Almost every crackseed store (Hawaiian candy store) has it.
Personally, I have never had bubble tea from its Taiwanese birth place. So i cannot attest to its likely awesomeness. But in my opinion, the best bubble tea is from Hawaii. (I admit I am extremely biased, but it is true.)