In the botanical sense, a fruit is derived from a flowering plant, mainly from the ovaries, and is usually the means by which a plant spreads itself (ie, a fruit contains the seeds of future plants).

When we talk about fruit in the food sense, we typically mean a sweet fruit that can be eaten uncooked. However, because of the botanical term things sometimes get classified as fruits that we would usually think of as vegetables and vice versa - for example, the tomato, with its seeds etc, is technically a fruit, while rhubarb is actually a vegetable.

Generally, when you talk about a vegetable you mean the leaves, root or stem of a plant, and not the seed part. Nuts are technically fruits, and so are the pods of beans etc.

You can find fuller definitions and examples here.