For example, one of these funny-looking fellows?
You can eat them raw or cooked (it's best to avoid eating the skin - although I have seen recipes where it's diced and cooked with the skin on - as it can be quite tough).
You need to either peel it, and then cube, julienne, slice or shred etc. as required for your recipe; or you can halve it, then bake, boil or steam etc. until it's tender and scoop out the flesh. In the middle, there is a flat stone, which is also edible, although it is slightly bitter tasting, especially the outer part of it.
The flesh, when raw, is crisp like an asian pear, with a taste reminiscent of a cross between raw courgette, melon and a hint of pear. It's a pleasant surprise, as they are billed as being 'bland', which I would disagree with (well, unless you think the taste of courgette/melon is bland). I find it quite pleasant, and will definitely be incorporating it raw, perhaps julienned or shredded/grated into salad/coleslaw/garnishes etc. especially with that hint of melon/fruitiness to the flesh. I think it will go well with seafood and chicken.
They are also popular baked and stuffed with a mixture of their own flesh and other ingredients (Louisiana style, with prawns), and would be great in casseroles and stews as well as soups.
You could substitute courgettes (zucchini) for chayote (they would need less cooking time though) in a recipe, whether raw or cooked. Courgettes wouldn't need peeling.
Chayotes are 19 calories per 100g, and you can find more information about them here.
If you'd like a recipe to try one out in, you could have a go at my Spicy Mirliton and Shrimp Soup, a popular Louisiana dish. (Or have a go anyway, and use courgette if you can't get hold of one!)