Most glasses can be categorized as tumblers or stemware. Stems were added as handles to prevent the temperature of your hand from affecting the temperature of your drink. The shape can make a difference as well, but since I have no desire to become a wine afficianado, I'm not going to bother figuring it out.* If you want to increase your knowledge further, this site was quite helpful.
During our multi-month experiment, we used (or will use) the following glasses:
large wine glass
tall tulip glass
I didn't have to buy the last two; apparently we've been using highball glasses as water glasses and an old-fashioned glass for our morning milk as long as I can remember. I also learned that our crystal goblets (used for water for Sunday dinner in the dining room) are, indeed, water goblets and not wine glasses. (See here.) Wine glasses are thin and smooth and water goblets can be thick and etched. Did you know that? I was thinking I could use them instead of buying a tall tulip glass, but apparently not!
*Here's a sample of what you could learn: "A white wine glass is small, designed to keep the wine cooler longer. White wines have a less intense aroma, and their glasses have smaller mouths and bowls because they don't need the extra surface area for the best flavor and aroma as red wines do."