A jillion is an enormous number of something. If your next door neighbor has too many cats to count, go ahead and say they've got a jillion cats.
Similar to words like zillion, tons, or oodles, jillion is perfect for talking about a huge but vague number. It's also hyperbolic — in other words, it's an exaggeration. The word is modeled on actual numbers like million and billion, so it almost sounds like a real quantity. But like zillion, jillion is imprecise. Its origin is vague too, described as an "arbitrary coinage" first used around 1940.