What kinds of cows have brown spots?

Hereford was the first answer given. While Herefords are probably the most well known, at least in the United States, there are several breeds which are brown (actually the color of some is called "red") and white. The markings of some are similar to those of the Hereford, but some are much different.

The Australian Braford and the Beefmaker, which look very similar to Herefords, are breeds derived from a cross of Herefords with other breeds. Others breeds which do not resemble the Hereford in appearance include the English Longhorn, the Texas Longhorn, and the Simmental.

There are breeds which have a variation in color, with some being black and white while others are red and white.

For a fairly complete listing of cattle breeds with pictures of most, see Oklahoma State University's "Breeds of Cattle" page.
A brown and white cow is the markings of a particular breed of cattle. The most popular breed that is brown and white in the USA is the Guernsey. They are primarily used as a milk cow and produce milk that is medium high in butterfat content which can be used to make butter, ice cream, cheese, etc. as well as milk to drink.

Brown-and-white cattle also come in other breeds, like Ayrshires, and the old-style Simmental-Fleckvieh cattle.