How do I know if a food or recipe can be made in a microwave oven?
Microwaves in general provide all-around, not too powerful heat. Very roughly, this means it's kinda like cooking on a low-powered stove, except the heat comes from all directions, not just below. (And depending on the microwave, it might heat a good bit less from below.) The things that work are therefore the things that work with relatively gentle, constant, extended heating. This translates approximately into the following traditional cooking methods:


Steaming. If there's not a whole lot of liquid, and there's some empty space for steam to move around in, this is what you'll get. For example, a covered bowl of vegetables (maybe with a little extra water), or even fish on a plate covered with plastic wrap, will come out about like if you steamed it.
Heating, simmering, boiling. If you've got a more substantial mass of food, you'll get something on the continuum between these three, depending on how much of it there is, what power you set your microwave to, and how long you cook it. This means you can do anything from melting butter (carefully) to cooking oatmeal to boiling noodles. (And along with that, you get the canonical use: reheating food.)
Dehydrating. This is a bit of a corner case, but if there's relatively little water in something to begin with, and it can escape easily, the heating will dry the food out substantially - this allows things like dehydrating herbs or even microwave jerky.


And the things you can't do? Anything that requires high heat, especially dry heat, anything where you'd expect the food to brown or get crispy. Roasting, broiling, pan-frying, and so on - the heat from the microwave just isn't going to accomplish it. And on top of this, anything where the cooking time really matters is a bad idea. Even if the microwave might eventually get it hot enough, it's going to take a bit to get there, so if you're looking to quickly cook something in an already hot pan then take it out, it's not going to work. Yes, there are complex ways to work around some of this (for example the herb-frying Brendan mentioned) but as far as what's straightforward to do, you'll want to avoid all these things; they're easier to do the normal way.