The term "puck" or "pooka" had been used for some time prior to the Bard's time to refer to mischievous fairies or devils, but Shakespeare's use of "Puck" to refer specifically to the name of this one particular fairy pretty much solidified the use of Puck as a proper name in popular culture. Within the play, he's the jester at Oberon's court. "Robin Goodfellow" is sort of a nickname, which most likely came from the tradition of not referring to fairies by their proper names, so as not to attract their attention. The euphemistic names usually included some reference to the fay as "good" or "fair," even though they were really often quite nasty, presumably so as not to get their dander up. Puck himself was held responsible for many acts of mischief, including changing his own form to deceive travellers, and making milk sour. He was also occasionally said to do helpful tasks, however, like sewing and housecleaning, and people would sometimes leave out milk for him. In such a role, he's basically a variation on all the stories of hobgoblins, kobolds, and other house-spirits who perform household tasks.
The legendary bandit Robin Hood (appropriate for a mention today, I suppose, seeing as how it's Errol Flynn's birthday) is sometimes viewed as a humanized version of Robin Goodfellow. Robin Hood doesn't have any magical powers, but he DOES live in the forest, dress in green, and have a sense of mischief. Wikipedia says that this is unlikely to have been the source of the outlaw's origins, but it's still an interesting thought.