To taste wine, sip and hold it in your mouth. Different parts of the tongue register different tastes. Allow the wine to roll all over your tongue, and notice the texture, or mouth-feel, of the wine. Balanced wines represent the harmony among several components: aroma, acid, tannin, fruit and sweetness. Acidity should provide a pleasant, but not overwhelming tartness. Balanced tannins contribute an agreeable astringency (that slight "pucker" feeling); this is one reason many red wines pair well with fat-rich foods — tannins in red wines cut through the fatty mouth-coating after each bite and prepare your taste buds to enjoy the next bite anew.
Finally, what kind of taste does the wine leave in your mouth after you have swallowed, and how long does the taste last? This is the wine's finish. A clean, crisp yet lingering balanced finish is the mark of a good quality wine.