Cyberpunk as a genre is rather hard to pin down. It emerged as a twist on other forms of sci-fi, and then was it self-twisted and changed so quickly that many feel cyberpunk is already a dead form. It doesn't help that it's tied into "punk", already a difficult and much debated genre of its own.
At the core, you can get a decent but pithy understanding of the genre with any of numerous little phrases, some already mentioned. Cyberpunk can be "a science fiction genre noted for its focus on 'high tech and low life.'" Or it's film noir with emphasis on information science. Or it's sci-fi twisted so that technology is a tool of classist oppression rather than an enabler of new developments. The atmosphere of paranoia and surveillance from Orwell's 1984 are often used in cyberpunk, but not many would classify that novel as being particularly cyberpunk.
Neal Stephenson's books are themselves odd contenders in this regard. Snow Crash is clearly cyberpunk, but compare Hiro to Neuromancer protagonist Henry Case. Despite both being hackers, Case is much closer to the "nerdy lowlife" we might expect in such a novel, while Hiro sets the standard for future protagonists like Neo in the Matrix: swordfighter, sunglasses, always cool and collected. The Diamond Age presents many cyberpunk themes — technology, computers, classism, unlikely heroes — but focuses on nanotech rather than cybertech. So much so that many consider it one of the first post-cyberpunk novels. Even his later books, like Cryptonomicon, hit many of the cyberpunk tropes without ever veering close enough to earn that genre — prompting some to dub it "cypherpunk."
Sadly, not even games are safe. Released in 1994, System Shock is the seminal cyberpunk game, and is true to its genre. But by 1999 and System Shock 2, we're already diverting into new territory with the increased emphasis on biological modification and the nature of intelligence. SS2 can easily be seen as a battleground between the ideas of two genres, the aging cyberpunk and the emerging biopunk. This culminates when many of the original Looking Glass team go on to make Bioshock, a game so similar in structure to System Shock 2 it could be called a remake wearing a new theme. Same story, new metaphors.
So at last maybe we can get at the root of what cyber-punk really is. On the one hand, you need punk. The surface visuals of this add much of the "street lowlife" element: mohawks, leather jackets, and invasive body mods (remember, cyberarms are just an extreme form of tattoos and piercings). The thematic underlying elements of punk are rebellion, classism, class uprising, and "grassroots movement." Stephenson used open-source software and hackers for this. In Deus Ex, note that the hero starts out working for "the man" but is eventually swayed to the rebellion. The same basic story is played out in Johnny Mnemonic as well.
On the other hand, you need cyber. The way that technology empowers rebellion, enables communication, breaks censorship, and changes people are big themes here. "Jacking into cyberspace" and bodymods are visceral ways to show that the people take technology (traditionally a tool of the empowered class) and turn it around, merge with it, and make it work for them. They subvert the system from inside it. If you subvert the cyber and replace it with some other metaphor, you get some other kind of *-punk (biopunk, nanopunk, but oddly steampunk seems to have gain enough momentum to morph into something else entirely its own).
All the other elements commonly attributed are taken from the "parent genre" that the author is working the cyberpunk into. Gibson wrote noir, the Matrix is an action movie, and System Shock is an RPG. It's easy to mix up the cyberpunk and the other stuff. Especially when multiple examples exist. If your primary exposure to cyberpunk is Blade Runner and Neuromancer, the idea that noir detective stories are bound to cyberpunk is easy to walk away with. But I think I've managed to separate out the core themes for you and provide a few examples along the way. Hopefully, that's a start.