Nowhere Boy takes a look a John Lennon's life from 1946-1960 and his relationship with his mother and his Aunt Mimi, who raised him. We also briefly see John's Uncle George, who had a huge influence on molding Lennon's sense of humor. Aaron Taylor-Johnson looked somewhat like John (or at least like he could be Julian's brother), and that resemblance and being able to copy Lennon's speech cadence helped make his role believable.
Nowhere Boy doesn't travel into Beatles territory, although there are subtle references in the film to events/places that appear later in Beatles' songs. As far as the music part of the film, it shows Lennon forming The Quarrymen, how he met Paul McCartney and Paul's friend, George Harrison, and having them join his band. At the end of the movie, John leaves for Hamburg and this is as close to The Beatles as the film gets.
As with any biopic, I'm sure there are probably inaccuracies in Nowhere Boy, but I'll leave those to the Beatles' experts to point out. I will say, I thought Thomas Brodie-Sangster was miscast as Paul, he appeared too young and too short of stature to play Paul. I came away from the movie with a better understanding of what made John Lennon the person he became as an adult. Even if you're not interested in John Lennon, you might still like Nowhere Boy, since it is an excellent family drama, one that just happens to be about a famous person.