When Luffy was young he met a band of pirates led by a kindly pirate named Shanks. Luffy befriended that band of pirates and from that moment on wanted to become a pirate as well.
During his time spent with Shanks, Luffy ate a Devils Fruit and unwittingly turned himself into a Rubber man capable stretching, inflating, or bouncing back impact based attacks. Because Luffy didn't know the nature of the fruit they remained friends, but after that Luffy seemed to be even more defensive of Shanks seemingly meek band of pirates.
After Luffy stood up to a local bandit that insulted Shanks and his crew, Luffy got caught up in a hostage situation and was later rescued by Shanks. However the incident cost Shanks his arm.
Shortly after Shanks set sail along with his crew, but before leaving he made Luffy promise that one day he would become a grand pirate and left his Straw Hat as a parting gift.
Luffy has since worked diligently on keeping that promise and is known famously as "The Straw Hat Pirate." Luffy now aims not only to be a grand pirate, but to become The Pirate King. Most people don't take that goal of his seriously, until they see his absurd strength.