Burger King, often abbreviated as BK, is a global chain of fast-food restaurants serving burgers, fries and shakes. Burger King is headquartered in Miami, Florida.
Peter, his mother Thelma Griffin, and her then-boyfriend Tom Tucker went to a Burger King in "Mother Tucker". The cashier thought Peter, who was going through a childish behavior stage with a fatherly figure now present, was adorable.
It's also mentioned in "Road to Rupert" when Peter points out that he doesn't go there to tell Lois what to do when she criticizes his building a ramp to jump cars.
In "Cool Hand Peter", it's parodied as Burger Queen.