The short version
A generative model is a system trained on a very large amount of text, images or sound, which learns the statistical shape of that material well enough to produce new material of the same shape. When you type a question, it is not looking anything up. It is producing the continuation that fits the pattern of your input.
That single fact explains almost every strange behaviour people run into. It explains why the output is fluent even when it is wrong: fluency is exactly what was learned. It explains why it can be confidently specific about something that does not exist: a plausible-looking citation has the same shape as a real one. And it explains why asking the same question twice can give two different answers.
Where the vocabulary misleads
Saying a model knows, thinks, understands or hallucinates imports a mental picture that does not fit. The most damaging of these is hallucination, because it suggests a malfunction, an occasional glitch in an otherwise reliable memory. There is no memory to glitch. Producing text that fits the pattern is the normal operation, and it does not distinguish between cases where the pattern happens to match reality and cases where it does not.
Once you hold that, the practical rule follows: the model is useful where fluent, well-shaped output is the goal, and it needs a check wherever being factually right is the goal.
What the terms actually mean
A model is the trained system. A prompt is your input. A token is the fragment of text it works in, roughly a short word or part of one. Context is how much it can take into account at once. An agent is a model wired to take actions rather than only return text, which multiplies both what it can do and what it can get wrong.