This is an important and difficult question and there isn't a settled 'right answer' about it.
One thing that is clear is that conceivability should be distinguished from "imaginability". I can conceive of what a thousand sided regular polygon would be quite clearly. I know exactly what that phrase would refer to and I could use my geometrical knowledge to prove things about such a figure. However, I can't really clearly imagine what it looks like. It'd look a lot like a circle form some distance, but I can't picture in my mind the difference between the regular 1000-gon and the regular 999-gon or the regular 1001-gon.
Another thing that is clear is the conceivability should be something like being able to understand or grasp some concept. The things that are supposed to be inconceivable are contradictions like "round square." And that much seems right--every contradiction is inconceivable. The harder question is whether every conceivable situation isn't contradictory. Is it conceivable that Goldbach's conjecture is true, or that it is false? It's an arithmetic conjecture, so it is either necessarily true or necessarily false. And if it's necessarily false, then it's a contradiction in terms, like a round square.
People who are skeptical about the use of conceivability as a guide to logical possibility think that examples like the goldbach conjecture show that "conceivability" is just lack of knowledge of the contradiction. Maybe people before modern geometry would have found it "inconceivable" that there be triangles whose interior angles added up to more than 180 degrees. But they were wrong about what is actually logically possible, because we can clearly see now that there are hyperbolic geometries, and so forth.
People who do like to use conceivability as a guide to logical possibility, on the other hand, agree that such cases point to a difficulty for their theory--because it means that people pre-Riemann only thought that something was inconceivable when in fact it wasn't. However, they say, there simply isn't any other alternative. What else is the guide to logical possibility, if not conceivability?
That's roughly the state of play today.