Well, Dreamcast nailed the rest of the stuff, so let me clarify on this. Generally, demons have one of two types of events that'll happen. One is a question. The answer to this is based off their personality, so there's an element of luck unless you REALLY know your demons well. But you can get a general idea of what to answer based upon the demon. The other is a request for something. Usually, after two or three events, you'll get the option to stop negotiating with that demon via an option called "End Talks." For many demons, this will usually get them to join you. Some demons will join you right after the first question if you luck out/correctly select the right answer tho.
In general, just know when to stop negotiating with a demon. If you don't end the talk, you'll have a much harder time convincing them to join. Also, another reason why you want to recruit as many demons as possible is because negotiating with a demon that's already in your party has many benefits. You could be healed HP/SP, you could get an item, or money, etc. In all cases, it'll also end the battle immediately which is a FANTASTIC way to not get slaughtered if you run into a bad scenario.