Considering that the Goth subculture began with punk, it's hardly surprising that many Goths (particularly eldergoths and old-school or trad Goths) still enjoy listening to the subculture's musical roots. Punk, or more correctly punk rock, developed in the 70s and was characterised by short songs, fast, stripped-down music and controversial, often political lyrics covering anti-authoritarian themes. There are also dozens of subgenres of punk including Oi, hardcore, surf punk, Celtic punk (e.g. Flogging Molly, whom I adore), crust punk, folk punk, horror punk, riot grrl, ska punk, gypsy punk (e.g Gogol Bordello) and streetpunk, and punk has had a strong influence on many other genres - including Gothic rock.
Please note: punk and pop-punk are not the same. Most punks and Goths would be amused or downright offended if you suggested that their primary musical tastes covered bands such as Blink 182, Good Charlotte and The Offspring. Pop-punk is generally neither dark nor Goth-friendly, although I'm sure I'm not the only Goth who has occasionally been known to listen to one or two Green Day songs; and many mallgoths have shown a fondness for the likes of Good Charlotte.
Where was I? Oh, yes. Instruments typically used in punk music include electric guitars, an electric bass, vocals and a drum kit, nine point nine times out of ten eschewing fancy techniques and expensive special effects. Lyrics are often shouted rather than sung.
Punk and its various subgenres are considered Goth-friendly because of the influence it had on the Goth subculture, and the DIY ethic and non-conformist spirit that is inherent in both punk music and the punk subculture.
Punk bands include: The Sex Pistols, The Ramones, The Clash, The Adicts, The Gun Club (whose members included Patricia Morrison, later of The Sisters of Mercy, Fur Bible and The Damned, and Kid Congo Powers, later of Nick Cave and the Bad Seeds), The 4-Skins, The Jam, Motion City Soundtrack, Rancid and Vice Squad.
Protopunk is a term that has been retrospectively applied to bands and musicians who were influential in the development of punk rock. Some of these bands are popular in the Goth scene and can be heard at Goth clubs all over the globe. Despite all being tagged 'protopunk', these bands do not come from a single specific genre.
Protopunk bands include: The Velvet Underground, Roxy Music (who were, incidentally, the first band I ever saw live), David Bowie, Iggy Pop (and of course The Stooges), Alice Cooper (the third band I ever saw!), Patti Smith, David Peel, The Who, New York Dolls, Doctors of Madness.
Riot Grrrl began as an underground feminist punk movement, whose bands' lyrics address topics such as rape, domestic abuse and female empowerment. It continues today as an underground subculture and movement, with strong political and (obviously) feminist views and the typical rebellious DIY ethos. Female musicians that originally inspired the Riot Grrrl ethos included Lydia Lunch and Siouxsie Sioux. Jack Off Jill and Hole have also been somewhat dubiously associated with Riot Grrrl on Wikipedia; Jack Off Jill enjoy moderate popularity in the Goth subculture.
Riot Grrrl bands include: Bikini Kill, Bratmobile, Heavens to Betsy, Calamity Jane, Excuse 17, Julie Ruin, Lunachicks, Tattle Tale, Voodoo Queens and Huggy Bear.
Emo as a music subgenre originated in 1980s America as a development of the hardcore punk movement. Nowadays it is a much-maligned subculture, often wrongly confused with Goth, and has spawned other subgenres such as scene, which is about as dark and Goth-friendly as most pop-punk music. I would suspect that most scene fans are relatively unaware of their culture's roots in punk.
However. Emo is generally characterised by moody and introspective lyrics and melodic musicianship, untypical of a lot of music that has its roots in punk rock. It was originally known as 'emotional hardcore' or 'emocore'. Modern emo music has been combined with indie rock, creating bands such as Panic! At The Disco and 30 Seconds To Mars.
Most crossover between the Goth and emo subcultures is fashion-based rather than musical, but because of its expressive lyrics and gloomy atmosphere, its punk roots and indie associations (independent music is, of course, a Good Thing to most Goths), I figured it was Goth-friendly enough to be included here.
Emotional (or emotive) hardcore bands include: Embrace and Rites of Spring.
Emo bands include: Dashboard Confessional, Hawthorne Heights, Jawbreaker, Matchbook Romance, The Used, Taking Back Sunday and Sunny Day Real Estate.
Apparently, as I am informed by commentor Becky, the difference between the two lists above is similar to the relations between Goth and mallgoth (and yes, I have seen fans of the latter list of groups referred to online as 'mall emos').
Horror punk (aka horror rock) blends horror movie themes with influences from early punk and sometimes rockabilly. The lyrics often express black humour and are generally non-political, although there are some exceptions. There are also a few, lesser-known subgenres such as horror hardcore (e.g. Samhain, Septic Death) and horror metal (e.g. Ripper, Necrophagia). It often has a heavier, more metal-oriented sound than traditional punk.
Horror punk has become associated with Goth via its dark and sometimes cheesily humourous B-movie imagery, and with the popularity amongst Goths of bands such as Misfits.