You've seen them in import shops and Japanese restaurants: little cat statues, usually white with spots, happily holding a paw up to one ear. They are examples of the maneki neko, the beckoning cat, a common good-luck charm in Japan. Left paw raised = beckoning people closer; right paw raised = attracting wealth. They are always appealing objects, and so popular that you think, well, it might just work. I've had one or two in my time.
Though like most luck charms its origins are obscure, you'll see the most maneki neko figures at Japan's Gotokuji Temple, where a beckoning cat was said to save a feudal lord from mishap and thus brought great fortune to its temple home. Here's a story about that with some commentary! And here is a page from a jolly blog with photos of the temple.
There's all different kinds of maneki nekos: this page, from the absolutely fabulous website of a collector, lays it out for you.