Falstaff was originally named "Oldcastle," but the descendants of the historical Sir John Oldcastle pitched a fit. So, Shakespeare had to change the name of his fat, disgraceful knight to "Falstaff." The remnants of the character's original name can be found in the play when Prince Hal calls his pal "my old lad of the castle" (1.2.5). (Source: Falstaff, ed. Harold Bloom, New York: Chelsea House (1992).)
Some literary critics, like Stephen Greenblatt, believe Shakespeare may have based Falstaff's character on fellow writer Robert Greene (a grouchy man who bagged on Shakespeare in a pamphlet called Greene's Groats-Worth of Wit). Robert Greene famously referred to Shakespeare as an "upstart crow" – that's Elizabethan for "wanna-be," "poser," and so forth. Greene (1558-1592) was dead by the time Shakespeare wrote Henry IV Part 1 (c. 1597) but we like to think he would have been flattered if Shakespeare did in fact use him as a character model. (Source: Will in the World, Stephen Greenblatt, New York: Norton, 2004.)
Shakespeare coined the phrase "send him packing" (along with about a gazillion others), which appears in print for the very first time in Act 1, Scene 4 of Henry IV Part 1. After the tavern hostess announces there's a messenger at the door to see the prince, Falstaff says "'Faith, and I'll send him packing," which means he'll tell the guy to scram, vamoose, get lost, beat it, etc. (Source)
Giuseppe Verdi's opera, Falstaff, is based on the antics of, you guessed it, Sir John Falstaff. Mistress Quickly and Bardolph (whose cool Italian name is "Bardolfo") also appear. Check it out on YouTube. (Source)
Falstaff Brewing Corporation named their beer for Shakespeare's larger than life character. (Source)
In the hip-hop musical "Clay," the relationship between "Sir John" and rapper wannabe "Clifford" (a.k.a. "Clay") is loosely based on the relationship between Prince Hal and Falstaff in Henry IV Part 1. Check out The New York Times Theater review of "Clay."