Finish: the lingering flavour and sensation of the wine after drinking. It can be short, long, dry, complex…you get the idea.
Nose: the overall smell or aroma of the wine.
Sugar scale: a numerical system used to describe how dry or sweet a wine tastes, with 0 being the most dry. This system is being phased out and replaced with the more vague qualitative categories of extra dry, dry, off-dry, medium sweet, and sweet.
From the LCBO website.
Tannins: more common in reds, tannins are what make your mouth pucker after taking a sip and make a wine taste more dry or bitter. Tannins come from contact with the grapes’ seeds, skins, stems, and sometimes wooden barrels that the wine is aged in.
Varietal: the type of grape (or grapes, if it's a blend) used in a wine. Different appellations are conducive to growing different kinds of grapes, which is why some areas are known for a certain type of wine.
Vintage: the year the grapes were harvested in. Not all wines specify the vintage, because sometimes they use grapes from different years and hope that no one will notice the difference.