See Wikipedia's entry: Western (genre). I'm sure they explain it infinitely better than I could. But basically, this category is for shows set in the American Old West (generally in the second half of the 19th century). I'm sure there are shows that might be considered Westerns on some level, which might be set a bit earlier or a bit later, or even in the present day. Though most of those set in the present (or even the future) would be more immediately of some other genre, like comedy or sci-fi. I dunno.
See also Western films