Why are cows black and white?

Actually, cows aren't always black and white! There are many different breeds of cows, all of which tend to have different coloring. The black and white cows you are probably thinking of are Holsteins, a particular breed of dairy cow. Other breeds of cow can have brown, red, and/or white coloration.
To address your question of why these cows are black and white, we'll have to go over some basic genetics. When two Holstein cows reproduce, they pass on some combination of their genes to their offspring, and the genes that appear in their offspring's DNA will determine what traits that offspring has, including its coloration.
The gene that determines coloration in cattle breeds like Holsteins has two alleles. Alleles are variations of the same gene that express themselves as different phenotypes (physical traits). One of these alleles—the one that produces black coloring—is dominant. The other allele—which produces red coloring—is recessive. If either of a cow's parents pass on a dominant "black" allele, the offspring will have black coloring. However, both of the cow's parents would have to pass on a recessive "red" allele in order for the offspring to have red coloring. This means there are fewer possible combinations of genes that can produce red coloring in the cow. As a result, black/white coloring in certain cattle breeds is much more likely and much more common than red/white.