Drama; Film Noir; Cautionary Tale
On its most basic level, The Lost Weekend's a drama. It deals with heavy subject matter, including alcoholism and infidelity, and the plot is deadly serious outside of the occasional bit of comic relief. (Thanks, Bim.)
Many critics also consider the film to be an example of "film noir," a cinema genre popular during the 1940's known for dark and lurid plots that often focused on seedy detectives and sultry femme fatales. (Think The Maltese Falcon.)
While The Lost Weekend doesn't feature these plot elements, it does have many commonalities with the genre as a whole, from its use of low lighting and shadows as a visual effect to its focus on controversial subject matter.
It's also worth noting that the movie features many aspects of a cautionary tale—a story that cautions its audience against some potential moral threat. Like Don's fictional novel The Bottle, The Lost Weekend is meant to show the horrors of alcoholism to its audience, hopefully preventing them from going down the same dark path as our protagonist.
This aspect of the film is most prominent in its back half, when Don witnesses the horrors of alcohol withdrawal firsthand—no one wants to see a mouse/bat battle, especially when it's a hallucinated mouse/bat battle.