For simplified (and entertaining) explanations, read the fifth amendment guide at lawcomic.net. And here's a handy flowchart.
They only must stop if you have also asked for your lawyer. The "right to an attorney" means "right to an attorney to be present at all interrogations." So they cannot continue to interrogate you after you've asked for an attorney. And "interrogate" does not mean "put you under a light back at the station and ask you questions." It means any action intended to provoke an incriminating response. So perhaps they could ramble on about the case and see if you react. And if you don't specifically invoke your right to silence, your silence can be used against you. See Salina vs. Texas.
If you've only invoked your right to silence but not an attorney, they're supposed to stop questioning you. If they don't, and you answer anything, those answers cannot be used against you during trial. However...if answers that you give them lead them to new evidence, that evidence can be used at trial. If you said "I invoke my right to remain silent." And they say, "sure, sure. But say, you wouldn't happen to know where the knife is?" and you foolishly say "Oh, yeah, I threw it in the dumpster," they cannot testify that you said you threw the knife in the dumpster. But they can go get the knife from the dumpster and present the knife itself as evidence. Also, if you tell them something (in response to questioning. That should have stopped, but didn't. Not something you voluntarily say) and then you say something different at trial, they can now introduce what you said to them here. On the stand you say "I never had a knife" and the cop can say "actually, he told us he threw the knife in the dumpster." You were either lying to the cops then or the court now, and you can't do either of those things. That is a different matter than the crime you were accused of.
Also, after time has passed, they can come back and start asking you questions about a different matter than whatever question you invoked your right to silence on before. And then you need to assert your right again.
So you have to clearly say you invoke your right to silence, AND that you want an attorney present, AND then shut your stupid mouth.
These ideas primarily involve times you are in police custody. Custody means "I can't just walk away." If you can just walk away...do it. This is about dealing with interrogations where you can't walk away.