Punk rock, as a genre, is decided to have started in the mid-70s. Although preceded by '60s bands such as MC5 and Iggy Pop and The Stooges, the genre truly began to take root around 1974, with the slow ascendancy of bands such as The Ramones, the New York Dolls, and Television in New York City.
The styles of these bands focused on a stripped down aesthetic and a rejection of establishment thinking for their respective decades and by subverting audience's expectations of music and performance. How they accomplished this varied widely from performer to performer. The Clash focused on political change, The Sex Pistols focused on generally disrupting modern ideas of propriety (as noted by their famed attempt to perform "God Save The Queen" on the Thames during Queen Elizabeth II's Silver Jubilee), and GG Allin focused on throwing his shit at people. Suicide (band) caused audiences to physically assault them simply by using a synthesizer.
Punk fashion was mostly popularized by Malcolm McLaren, manager of The Sex Pistols, and his girlfriend, fashion designer Vivienne Westwood. Initially based off of the styles of Television bassist Richard Hell (who McLaren managed at one point and became better known as the frontman for The Voidoids), it focuses on torn T-shirts and jeans, leather jackets and other industrious clothes, and unconventional use of everyday objects with a kinky subtext, such as leather cuffs and dog collars. Thanks to the popularized DIY aesthetic, the modification of such clothes with everyday objects (such as chains, spikes, and safety pins) as well as band patches also became popular.
Since punk was a subculture focused on disappointment with the establishment and upsetting the standard order, it met with quite a big backlash from agents of the establishment. Local news stations, especially in America, painted punk as a subculture typified by violence and drugs (Sid Vicious didn't exactly help matters here). One famous example of such a backlash was the Quincy episode "Next Stop, Nowhere," which is actually somewhat infamous amongst music scholars and punks for getting nearly everything wrong about the subculture, coining the term "Quincy punk" in the process. (Slightly less ridiculous but still laughable is the C.H.I.P.S. episode with punks vs. New Wave kids.) The tendencies actually carry on today; flip through your typical comic book, and odds are at least one member of a random group of thugs will have a mohawk or be wearing studded leather. Of course, naming your genre/movement/subculture after a slang word for juvenile delinquents kind of invites those sorts of stereotypes.
Anyway. Back to the history. Like any musical genre that's been around long enough, punk began to stratify around the '80s, when such subgenres as Hardcore Punk (focused on loud, fast, and abrasive music) and Oi! (otherwise known as street punk, focusing on the concerns of the working class and often appealing to skinheads) began to develop. By the mid-'80s, punk had kind of lost the cultural zeitgeist, and new genres, such as Post Punk (more personal exploration, less anarchy) and New Wave (more poppy, but still somewhat questioning of modern standards), began to take over.
In the late '80s, the Washington DC Hardcore scene spawned the Straight Edge movement. This started out with the marking of underage clubgoers with black X's so that they wouldn't be served alcohol, but became a subculture within the subculture of not drinking, smoking or doing drugs; casual sex and eating meat were also frowned upon. The lunatic fringe, Hardliners, went even further. In Europe, Vegan Straight Edge became a norm in the Post-hardcore scene during the nineties, alongside support for ecoterrorism and animal liberation. The black-and-green flag was most fervently waved by the US band Earth Crisis. Meanwhile, the DC scene started concentrating on more personal, emotional songs, and became Emotive Hardcore, or Emo for short.
It wasn't until Nirvana hit it big in the '90s that punk saw a resurgence, as bands such as Bad Religion, Green Day, and Rancid began coming to the forefront. This led to another problem for most punks, though; at the same time, bands such as Sum 41 and Good Charlotte were also emerging, who skewed more towards "pop" than "punk" on the "pop-punk" spectrum (yes, it's a subgenre) and represented a commercialization of the genre. Then Hot Topic emerged, and much bile was thrown, as it presented to kids the opportunity to become a rebel by... buying stuff at the mall and not ascribing to any certain philosophy. Hooray. Some social upheaval.
- Pop Punk, as typified by bands like Green Day, Screeching Weasel, The Ramones and Blink 182.
- Horror punk, as typified by The Misfits
- Street punk, very working class and 'laddish' as typified by GBH and (at times) Rancid. Street punk is also closely related to Oi!, a genre that is almost entirely separated by the speed of the music, with Oi! being the slower of the two. See also Skinheads.
- Celtic punk, as typified by Flogging Molly, The Pogues, and the Dropkick Murphys
- Folk punk, as typified by Against Me! or Andrew Jackson Jihad.
- Anarcho-Punk, like Dead Kennedys or Rudimentary Peni.
- Metalcore, combination of Punk and Heavy Metal sounds.
- Crossover thrash is an earlier version that used thrash metal instead of melodic death metal for the metal component of the sound. This largely extinct subgenre tends to be more popular with metalheads than metalcore.
- Riot Grrrl, feminist punk, typified by Bikini Kill.
- Orgcore, or Beardpunk, is a relatively newer genre that is basically gravely vocals over punk combined with elements of post-hardcore. Typified by bands like Hot Water Music and The Lawrence Arms.
- Crust Punk, which contains elements of both anarcho-punk and extreme metal
These days, punk is mostly an underground genre, with indie and emo taking its place in the public eye. Punk is still split into its multitude of subgenres. Punk artists are also likely to cross over into other genres depending on their origins, as demonstrated by the Dropkick Murphys above. Reggae music in particular was a major influence for The Clash, Bad Brains, and later on Rancid. The punk scene embodies all sorts of philosophies and politics, ranging from anarchist to conservative to "I just want to jump around and have fun," but usually tends to the left. Punk style has mostly stayed the same, although the band patches have changed and sneakers seem to be more popular than combat boots these days.
Anyone interested in learning more about punk should read Please Kill Me, or England's Dreaming for a specifically UK-centric history.