The following list points out some of the top buys in Europe (items on which a country really shines are in bold). Because taste in shopping varies, I've included all the sorts of special-to-that-country merchandise. If you want my advice on what to buy, though, stick with the hand-crafted and the unique.
Now "hand-crafted" can apply as much to a fine wine, hand-tatted lace, or crystal ware as to a Bavarian carved wood Nutcracker, Irish wool sweater, or painted Italian ceramic plate—so long as it's something you couldn't find anywhere else and something that to you helps define the local culture in some way or remember your trip.
If you have the money, Europe's a great source for art, antiques, and high fashion. The rest of us can have fun haggling for semi-antiques and leather jackets at street markets.
Antiques (especially 19th-century Biedermeir); gold, silver, and enamel jewelry; petit point embroidery; porcelain (Wiener Augarten); wool products (Loden).
Bohemian crystal, antiquarian books and prints, antiques, handicrafts, jewelry, Soviet-era kitschy trinkets (watches and so on with the hammer and sickle).
Antiques (Chippendale), aromatherapy (The Body Shop is British), fine ceramics (Wedgwood, Spode, bone china), Royal Family memorabilia, raincoats (Barbour, Burberrys), "tapestry" (needlepoint kits), tweeds.
Antiques, cosmetics, crystal and glass (Baccarat, Lalique, Saint-Louis), Faience ceramics, fine foods (pâtés, truffles, Dijon mustard), Hermés scarves and ties, high fashion, lace, lingerie, perfume, wine and champagne.
Beer steins, cutlery (WMF), handicrafts and wood carving (cuckoo clocks, nutcracker soldiers, toys).
Antiques, ceramics and cookware, crafts, embroidered clothing (vests, blouses), jewelry (especially silver), leather (more sturdy products, like bags and sandals, than supple items like clothing), flokati woven wool rugs.
Crafts, hand-knit sweaters (especially from the Aran Islands), lace, linen, recordings of traditional Irish music, Waterford crystal, woolens and tweeds (especially from Donegal)
Antiques, art, ceramics (folk and fine; from Faenza near Bologna, Deruta and Gubbio in Umbria, Vietri sul Mare on the Amalfi Coast, Caltagirone in Sicily, Grottaglie in Apulia), glass (Venice), fashion (Milan, Florence), industrial design (coffeepots, lamps, and so on), jewelry (gold and silver), lace (Venice—well, Burano, technically), leather (Florence), religious objects (Rome), shoes.
Antiques, cheese, chocolates, crystal (from Leerdam or Maastricht), Delftware ceramics, diamonds, flower bulbs (check with U.S. customs about which kinds you can take home), pewter (from Tiel).
Plaids and tartans, Scotch (the whisky, not the tape), sheepskins, silver jewelry, tweeds, wool sweaters.
Ceramics and tiles, lace, leather (Barcelona), sherry.
Cheese, chocolate, music boxes, tin- and copperware, toys, watches and clocks, wood carvings.