I would say it is a little bit like "a nice after taste." If it's a smooth after taste, it certainly isn't awful. Think of your favorite liquor. If it's cheap and not high quality, you're more than likely to have an unpleasant flavor in your mouth after you swallow. Typically, it's bitter, sour, or you feel like you could breathe fire. It's the kind of flavor that makes you want to squish up your face after you drink. In wine, it could leave your throat and tongue feeling dry. But if it's a high quality liquor, it will have a smooth after taste that is pleasant and makes your drink taste even better even though you already swallowed. It might be sweet, savory, or even slightly bitter, but it is never overpowering. Some people also use the term "smooth aftertaste" if it has no after taste at all. You may not even know that your drink is alcoholic if it is very smooth.

No. It means "smooth". Surely you know what a smooth flavour is on the tongue.

Sometimes, after swallowing something (like wine), there is a lingering flavor. Sometimes this flavor is similar and sometimes somewhat different than the main flavor one experiences while eating or drinking the substance. This is known as the "aftertaste".