How does cable television work? Specifically, the different channels showing different content in terms of violence, bad language, nudity etc.

The FCC and the advertisers.

Cable tv doesn’t have to abide by the FCC obscenity rules. The network channels (CW, CBS, ABC, NBC, Fox, PBS) do. But FX, USA, CNN, Comedy Central, AMC… all of those don’t have to. Now, “standard” cable channels usually self-impose some regulations, and then remove them as it gets later into the night. But HBO, Showtime, all those other premium channels – they usually have less standards to begin with (after all, why pay for those channels if you can’t see some skin and hear the f word??) and then remove them even more as it gets later.

HBO is sent over wires only to people who want it, so they can show whatever they want.
ABC is broadcast over the air that anyone can receive, so they can’t show nudity or obscene language.

Essentially, you pay additionally for HBO and those premium channels so anything goes. Basic cable is just that. And then there are rules for what you can show at what times of the day etc.

Some cable channels are free, but they still have less restrictive censorship because you have to choose to get them delivered to you.