There are lots of ways to cook eggplant. This is one of the most versatile vegetables.
You can cut them into thick slices and fry them. Alternatively, you can brush the slices with oil and grill them.
They are very good in stews. Examples of stewed dishes with eggplant are French ratatouille, Sichuan fish-flavoured braised eggplant, South Indian fish curry with eggplant.
There's a fantastic Japanese eggplant dish where the eggplant is split in half, spread with miso paste and baked.
You can make "lasagne"-like baked dishes with them. kyleanderson has given you one suggestion.
Then there's an eggplant souffle that I sometimes make. You grill slices of eggplant, then puree them when cooked and add them to the souffle base. Wonderful with a roasted capsicum sauce.
You can also roast a whole eggplant over open flame until the skin is completely charred. Remove the charred skin and you have lovely soft, smoky eggplant flesh. Blend the flesh with tahini (sesame seed paste), garlic and lemon juice to make baba gannooj, the Middle Eastern eggplant dip. Or you could leave out the tahini, in which case you would have an eggplant dip called batinjan m'tabal.
I'm sure you'll find lots of eggplant recipes on HubPages.