Is a tomato a fruit or a vegetable?

There are different ways to classify or categorize plants and plant structures.  We eat many plant parts including roots, stems, leaves, flowers, seeds, and fruit.  The root is the edible portion of carrots, radishes, beets, and parsnips. Asparagus is an example of a stem that is consumed. Plants grown for their edible leaves include lettuce and spinach.  The flower is the edible portion of broccoli and cauliflower.  Corn is an example of a seed that is eaten.  Edible fruits include melons, apples, pears, green beans, peppers, and many others. 

The term "fruit" has more than one definition. 
In common usage, a fruit is a plant part that is used chiefly in desserts. In contrast, a vegetable is defined as a plant part eaten with the principal part of the meal.  That plant part could be a leaf, stem, root, seed, flower, or fruit.  This makes snap beans, carrots, and spinach vegetables, while apples and cherries are fruit.  By this definition, the tomato is a vegetable in that it is usually consumed during the main meal.
Botanically, a fruit is defined as any plant part with seeds inside it that formed from a ripened ovary (or would have seeds inside of it if it were not a seedless variety). Therefore, apples, cucumbers, cherries, peppers, and watermelons are botanically defined as a fruit.  By this definition, the edible tomato is a fruit since it developed from a flower and has seeds inside it.

Therefore, the tomato is both a fruit and a vegetable, depending on which context (common usage vs. botanically) you are talking about.