Psychological Thriller, Film Noir, Slice-of-Life
Taxi Driver isn't a thriller like other thrillers: There's not a ticking time bomb hidden somewhere in the city… because the ticking time bomb is the main character. The suspense doesn't come from a chase or a spy battle. It comes from watching the main character descend into insanity.
We see how Travis' attempts to find love go horribly wrong, and we see how he reacts to that rejection and to his inability to navigate his environment emotionally. The "thrilling" stuff (which is really more like "chilling stuff") comes when we think Travis is going to go full-villain and shoot Palantine, and when he kills Sport and a couple of gangsters. We're left in a state of suspense when the movie ends, since we don't know whether he'll recover psychologically or not.
The movie's also a bit of a film noir, since it deals with crime in a dark, gritty urban environment… although there are no Maltese Falcon-style jaded detectives or moody femme fatales. As Travis drives around New York, and deals with criminals like Sport or with crazy people like the sick passenger who wants to kill his wife, he risks becoming more criminal and crazy himself. The city eats at his soul. There's lots of literal darkness in this movie, too: lots of shadows.
Finally, Taxi Driver's a slice-of-life movie, capturing some of the genuine reality of New York City in the 1970's. For a lot of people, the Big Apple simply was not a nice place. Crime was ballooning, it was economically downtrodden—if you weren't safely ensconced at the top of a skyscraper, you were scraping along, running the risk of encountering violence.
Taxi drivers, in particular, were frequently at risk, since their whole job involves picking up strangers. (This is reflected in the scene where Travis tells the other drivers how he heard another driver just got "cut up" by a madman.) Other movies from the time period, like the classic Midnight Cowboy, deal with the same New York underworld in an effective and moving way—in Midnight Cowboy's case, by telling the story of a guy attempting to be a gigolo rather than a taxi driver.