I'm from Buffalo and what the other posters said is somewhat correct.
The thing is that we primarily do this to our Chicken Wings and it's only become popular to do this to other types of chicken. It originated w/wings and has found a home in other forms of chicken.
Chicken wings are deep fried (no batter or breading) then pulled out of the fryer and placed into a large bucket that contains Franks brand hot sauce, melted butter, salt, pepper, and garlic. Some people add other spices -- but the recipe I gave you is what's most popular. You shake those wings around in the sauce and then serve 'em up w/a sides of Bleu Cheese dressing, celery and carrot sticks. You can get them hot, medium, mild or super-hot. Each restaurant has its variations on how hot "hot" is. For example, if you order hot in one place, it may be tame compared to other places.
Some places have started to do this to chicken fingers, chicken patties, etc. ... however, that's not "true" buffalo chicken. It's just treating regular chicken in the hot sauce. The only "true" Buffalo chicken is to add the word Wings after chicken.
They originated in Buffalo, NY at a restaurant called "The Anchor Bar". It's still around today and is one of the best places to go. Another popular spot to hit is Duff's. Personally, there's a little tavern near my school and I think they have excellent wings. Loads of places to go get them and everyone thinks their favorite is the best!