The first goth music grew out of punk, and some of the early bands were very lively, characterised by tribal drumming. Things changed in the mid-eighties, mainly due to the rise of the Sisters of Mercy, probably the most well-known goth band. The Sisters were characterised by deep vocals, simple yet prominent basslines, and simple drum machine rhythms. Unfortunately they came to typify the "Goth Sound" and too many later bands followed their example.
Later on, there was some crossover with the Industrial scene, and a lot of bands now use electronic sounds as well as the typical guitar/bass/drum machine/vocals (in fact some current "goth" bands sound suspiciously like old electro/industrial bands).
There is also a fair bit of "ambient" goth, characterised by subtle instrumentation and haunting female vocals.
Lyrically, goth tends to be on the doomy side- dark lyrics and a doomy sound are fairly typical of most (but not all) goth these days.