When milk coagulates it separates into two parts, a semisolid portion and a watery liquid. The liquid is whey and the semisolid is the curd. The curds and the whey represent the two basic groups of proteins found in milk. In order to separate them a coagulant must be introduced. Most cheese is separated with the coagulant rennin. Word to the wise vegetarian, Rennin is an enzyme found in the stomachs of calf’s that assists them in breaking down mothers milk. In the olden days rennin wasn't processed from the stomach so much as a portion of the calf’s stomach was finely chopped and then added to the milk. These days the rennin can be separated from the stomach chemically without adding any tasty meat bits to the cheese... but I digress. Left at this stage the cheese is called un-ripened or fresh. Popular cheese of this type includes ricotta and cottage cheese. To ripen, or age the cheese, a variety of techniques are applied depending on the desired result.
Spices or salt are added to arrive at the desired flavor. Heating, exposure to bacteria, both natural and unnatural, or soaking in a solution, completes curing. The consistency and grain of the cheese is determined by how much the curd was broken down before aging began. Large curds result in a smooth texture. Medium sized curds will make the cheese crumbly and small curds will give the cheese a fine grainy texture.
Then there's the pasta filata (spun paste), the curds are given a hot whey bath and then, when pliable, the curds are stretched and kneaded, much like taffy. Popular examples are mozzarella and provolone. The whey is then reheated until it coagulates again to make Italian ricotta. According to Mario Bitali, Italian cheese makers are required by law to make ricotta from their whey.
The curd most popular in the US is undoubtedly cottage cheese. Coming in second would be the curds identified in the previous write-ups and are the natural curds of un-ripened cheddar. The curds are separated from the whey and pressed. The result is a small bit of cheese with a subtle flavor that tends to squeak against the teeth. Because it is un-ripened, this type of curd tends to lose its desirable qualities and become quite dry and salty after only a few days. If you don't live in a dairy area chances are you won't be able to find any.
Incidentally, the curds and whey that little Miss Muffet was chowing on was probably cottage cheese or a close facsimile.