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A fruit is the mature ovary of a seed plant, usually developed from a flower. Fruits have seeds so they further the reproductive cycle. A vegetable is a plant or that part of a plant which is edible, and does not necessarily have a role in the plant's reproductive cycle.
A tomato is both a fruit and a vegetable, depending how (and why) you want to look at it. Botanically, it's the fruit of the tomato plant. For purposes of human consumption, though, it's primarily a vegetable. It's not generally eaten in the same way as a fruit like a banana or apple. It tends to be eaten more like a vegetable like onion or lettuce.