In botany, an aromatic refers to an oil that plants secrete. The compounds that were referred to as aromatics were originally investigated because many of them have a pleasant smell. Not all of these compounds, however, do.
All aromatics that plants have fit the definition of aromatic given above, that is, a benzene ring based structure. However, not all chemical aromatics are present in plants. Benzene, itself, for example, is not produced by any plant I know of.
Most aromatics are, of course, oil soluble due to the carbon ring structure. However, since many also have hydroxyl or corboxyl groups, some are also partly water soluble. The fact that most of these chemicals are relatively small with a single ring system and non-polar structure is what gives them their strong smell, because they diffuse into air very quickly. This is what seperates these essential oils from larger ring system fixed oils.
Most aromatics are or have been economically important for some reason or an other. Often this had to do with their aroma, such as in vanilla or peppermint, or sometimes to do with their medicinal value, as with aspirin.