Who could ever forget the famous scene where Robert DeNiro poses in a mirror with a magnum and asks "You talkin' to me?" You may forget it was from Taxi Driver, you may never have known it was from Taxi Driver, you may never even have heard of Taxi Driver, but you will not forget that quote.
The music, the main theme. It perfectly captures the stylish City Noir atmosphere associated with the film: existential, lonesome, and hypnotic. It was Bernard Herrmann's last work before he died during the production.
His motivations and methods were questionable but at the end of the day Travis got Iris reunited with her family and thats pretty awesome.