The basic definition of Gothic music is this: "Gothic rock (also referred to as Goth rock, or simply Goth) is a musical subgenre of alternative rock and post-punk that developed during the late 1970s." (Thank you, Google.) It combines dark music with introspective and usually sad lyrics. Often it features deep, droning vocals, e.g. Joy Division, The Sisters of Mercy.
Gothic music does not include ANY of the following bands:
etc., etc., etc. That doesn't mean that there is anything 'wrong' with liking any or all of the above bands - heaven knows I'm into most of them myself. But they are a far cry from what is considered Goth music, which is something you should really be aware of if you are interested in being a part of the subculture. Many of the bands on the above list are mistakenly labelled Goth by the media or general public based on their appearance, or because they have a 'heavy' or dark sound.
A note: because many metal and rock bands are also popular amongst Goths, it is not uncommon for them to be played at Goth clubs, be featured in Goth magazines or perform at Goth festivals. Many if not all Goths enjoy different types of music from lots of dark (and a few not-so-dark) genres. But true Goth music is the common thread of the subculture and the most popular genre within it.
As well as rock and metal, there are other, lesser-known subgenres that enjoy popularity with Goth music fans. Goth's closest musical relations are punk, Industrial (many events aimed at Goth club-goers feature a mix of Goth, Industrial and metal music), New Romantic, dark cabaret, neo-classical, dark ambient and steampunk. Other popular but less closely-related genres include J-rock (commonly associated with Visual Kei and Lolita fashion - more on this later), classical and ambient music, and Gothic, symphonic and operatic metal.
Also, as I mentioned before, Goth is an umbrella term for many sub-subgenres of music, including but certainly not limited to the following: post-punk, batcave, deathrock, darkwave, new wave, coldwave, dark or Gothic electro, and synthpop. It gets confusing, but with all these different types of music to choose from, the budding Goth is practically spoilt for choice.
To start checking out some Goth music, pop on over to Last.fm (don't worry, it's free).
Listening to: Meds - Placebo