There is some confusion that arises when talking about whether something is a fruit or vegetable, because there are actually different meanings of the words, depending on the context and who is using them.
In the scientific or botanical sense, a fruit is merely the part of a plant that contains the seeds. However, in the culinary sense, the word is only applied to those which have a sweet taste when raw. Botanical fruits which are not sweet are typically referred to as vegetables. An example would be tomatoes, which are technically fruits but commonly referred to as vegetables.
Since grapes both contain seeds and are sweet, they are fruits no matter how you look at it.