Ignoring the definition of someone who is just a "plain" goth, here are some major sub-divisions of the culture:
Cyber-Goth
A goth who crosses over between the goth and futurist (also known as "cyber"; see below) subculture. Cybergoths dress in futuristically inspired outfits (as opposed to the typically victorian influence outfits) and typically listen to darkwave (a combination of goth-rock and electronica), such as the Cruxshadows.
RomantiGoth
A goth who crosses over between the goth and New Romantic subcultures, or draws influence from either movement. Romantigoths dress in turn-of-the-century style wear, drawing strong influence from Victorian, Edwardian, Baroque, and even Renaissance fashion. Romantigoths are sometimes mockingly called "mopey goths". Romantigoths listen to music with such influences, such as Voltaire.
Industrial-Goth
A goth who crosses over between the goth and industrial subcultures. Industrial goths take on more abrasive fashion, drawing influence from rivet-head (industrial fans) fashion, as well as militaristic influence. Industrial-goths regularly listen to darkwave (see above) and EBM (a blend of industrial and techno), such as Nitzer Ebb.
Punk-Goth/Classic Goth
Goths devoted to the early scene (which grew out of punk, obviously). These goths tend to be older, and wear fashions of the time (tight black jeans, leather jackets, black aviators, Robert-Smith-hair, etc.). They likewise listen to early goth bands and proto-goth bands, like Adam Ant or Gloria Mundi.
Deathrock-Scene
Deathrock (not to be confused with death-metal) was a dark, ominous form of punk which grew alongside goth in the early 80's. Mostly develeoped in America by a handful of bands (most noteably Christian Death). The death-rock scene has wild, violent fashion, drawing influence from glam and punk, as well as from old horror movies. The scene was eventually absorbed into the goth culture.
Psychobilly
The odd blend of goth and rockabilly subcultures. Pioneered by such bands as the Meteors and the Cramps, the style of music has a small following, which has essentially been absorbed entirely into the goth-culture. Very similar to the death-rock scene, fashion wise.
New/Neo Romantics
The New Romantic movement was a fashion and philosohy movement of the late 70's and early 80's. Equipped with effeminate, glam-inspired fashion, the New Romantics (as the original romantics did) primarily believed that science and logic only made life worse, and that faith, love, and artistic expression would make for a better life.
Futurists
The Futurist movement was a musical and philosophical movement of the late 70's and early 80's. Musically, the futurists embraced progressed, experimental synthesized music. Philosophically (in opposition to the new romantics), Futurists believed in forward-thinking and innovation, and that these things would allow for new artistic and eccumenic expression.
Neo-Goth
"Neo-goths" as a whole aren't goths, but instead, are a new wave of club-goers who wear goth-influenced fashion. Neo-goth fashion is typical club fashion, though, with a darker twist. Neo-goths listen primarily to darkwave music, leaning more towards that which is more electronic in nature, such as VNV Nation.
GRaver
GRaver is a simple portmanteau of "goth raver". GRavers are much like neo-goths, and are, rather simply, ravers who draw inspiration from the goth culture.
EGA/EGL
This pair of acroymns stands for "Elegant Gothic Aristocrat" and "Elegant Gothic Lolita" respectively. Not so much an actual culture as it is a Japanese varient of goth fashion, followers of EGA and EGL will listen to darkwave as well as symphonic metal; the group Malice Mizer (who popularized the fashion) is particularly popular. The fashion revolves around a style which makes it's wearers look similar to Victorian porcelain dolls.