I would check out Williamsburg for good, innovative food in Brooklyn. We lived in Park Slope for 3 years, and I can't really recommend much in that neighborhood--or at least there was nothing that I would consider worth traveling from Manhattan to Brooklyn to eat. The best is probably Al Di La--really good Italian food (not red sauce Italian), located on 5th Ave and Garfield in Park Slope. Get there early, or you'll have a wait. Alternatively, you can eat in their attached wine bar (or just enjoy a glass or two while you wait for a table).
It all depends where you're staying, as Brooklyn isn't as easy to get around in as Manhattan. 14:41 made some good suggestions if you're in the Brooklyn Heights/DUMBO area.