Cocktails - like all fine things - have a culture, style, and traditional all their own, some of which is flair, but most of which comes from praticality and serving the best drink possible.
There are hundreds of styles of glasses, making it completely impractical for any home bar to stock all the options. Thankfully, that's a good thing - cause you only need five, which you can add as get excited about different styles of cocktails. The first three are absolute essentials, and the latter two are designed for specific classes of cocktails that are important to their enjoyment.
1. The Martini [or Cocktail] Glass: Traditional v-shaped glass that defines the martini, but could also be u- or egg shaped. A long stem keeps the hands away from the cocktail, keeping the liquids cold. Look for sizes around 4-6 ounces.
2. Rocks or Old-Fashioned Glass: Short glass, averaging around 8 oz. The key is to get one with a nice heavy bottom, which enables good muddling and crushing, and gives the glass - which will often only be one-quarter full of sprits - some heft.
3. Highball Glass: Tall, straight sided glass, averaging around 12 oz. Useful for summer-y drinks like a Tom Collins, fizzes, sours, and for non-mixed drinks, like soft drinks and ice teas.
4. Cordial Glass: Large bowl designed to explore the aromatic qualities of brandies, cognacs, and cordials. In the absence of other other options, this is a great way to serve most after dinner drinks, like the White Russian, etc. Plus, snifter is fun to say.
5. Flute: Standard for Champagne and other sparkling wines, designed to preserve the bubbles. Also makes a great way to serve other fruity and frozen drinks, like a Bellini.
As you expand your collection, try mugs for warm drinks, like an Irish Coffee or Hot Toddy mug, hurricanes, punch glasses, chimney glasses, and classy shot glasses.