When you're designing a cheese plate, you're looking to create a balance of flavors and textures. You can go all-goat, all-sheep, or all-cow if you'd like, but unless your crowd has highly specific tastes, you're best off aiming for variety.
Think about an ounce per cheese per person for appetizers or after dinner; a little more if you're serving less than 5 cheeses.
There are 5 major styles of cheese: Fresh, washed-rind, bloomy, pressed, and blue. Try to get an assortment of these, as well as some palate-cleansers (olives, nuts, bread) and flavor-enhancers (honey, quince paste, fruit preserves, or aged balsamic).
Fresh cheeses are soft, mild-tasting and creamy: Anything without a distinct rind is considered a fresh cheese. They range in flavor and texture from fresh mozzarella, ricotta, or goat cheese to salty brined feta.
Washed-rind cheeses are the funky ones: They're bathed in liquid, whether beer, wine, or water to encourage the development of helpful bacteria that add tons of complex flavor. Meunster's the most common washed-rind cheese available here; you might also see Taleggio or Raclette in your store. They're generally runny inside with semi-firm outsides.
Bloomy cheeses are rich and intense, with creamy insides and a semi-firm, edible white rind. Brie, traditionally made with raw milk (though, sadly, not in the USA), is a member of this family. Other favorites are Brillat-Savarin, Pierre Robert, or the super-runny St. Marcellin.
Pressed cheeses are firmer, with harder rinds. Most hard and semi-hard cheeses fall into this category, from Parmigiano-Reggiano, Romano, and Asiago, to Cheddar, Gouda, and Gruyere. (Gruyere is also washed with brine, but not to the same degree as the washed-rind cheeses) Their flavors are all over the map — Parmigiano-Reggiano is deep and nutty; aged Gouda is butterscotchy and young is creamy; and Cheddar can taste like anything, depending on its age.
Blue cheeses are funkily intense, shot through with veins of penicillin. They're usually creamy, sometimes with a salty crunch in spots. Stilton and Maytag Blue are classics, as is Roquefort, but check out Cabrales for truly complex flavor or mellow, nutty Fourme d'Ambert.