There are many gangster movies in the same vein as The Godfather, with similar themes and settings. Popular examples of this type of film include Goodfellas, Scarface, Carlito's Way, and Reservoir Dogs. You might also want to look at the work of actors such as Al Pacino and Robert De Niro, as both have extensive careers playing mobsters in classic gangster films.
There are also a lot of great gang films that don't have that classic mob feel to them. Some of my favourites include:
City of God (2002) – Based on a true story, this Brazilian film (also known as Cidade de Deus) is one of the best gang films I've seen. It's subtitled, but the film is so absorbing that you'll soon stop noticing.
La Haine (1995) – A black-and-white French film following the lives of three friends in their young twenties, living in the poor suburbs of Paris at the time of the riots.
Romeo + Juliet (1996) – Baz Lurhmann's modernisation of the Shakespeare classic is a surprisingly good gang film, considering it's based on a romance.
Lock, Stock and Two Smoking Barrels (1998) – Guy Ritchie is worth watching out for if you enjoy British gangster films - he also directed Snatch and Revolver.