Having children make butter from fresh cream is a popular demonstration for many elementary school teachers. It seems oddly unintuitive that a familiar solid substance could be made from a familiar liquid, just by shaking it. Nevertheless, that is exactly how butter comes into being. The primary difference between types of butter is how the cream is treated before churning.
The fat found in heavy cream is bound up in the form of globules, balls of fat contained within a delicate membrane. When the cream is agitated or whipped, the action of the whisk breaks open the globules and frees the fats, which are then drawn to each other. If the cream is agitated long enough the fats will form into a solid mass, floating in a puddle of thin, tangy milk. The remaining traces of milk are rinsed out of the butterfat in cold water and the mass is compressed into a solid lump or brick of butter.
Sweet Cream Butter
The familiar form of butter for most Americans and other nationalities of British origin is referred to as sweet cream butter. This is because the butter is made from cream that is sweet and fresh, rather than aged. The end result is butter with a light, uncomplicated flavor that is rich but mellow, well-suited for table use or baking.
France and some other European nations take a different approach to butter-making. In these countries, the cream is lightly soured by a bacterial culture similar to the one used in America to make buttermilk. When the cream is then churned into butter, it has a much more pronounced flavor, one Americans tend to identify as "cheesy." Cultured butter makes an excellent foil for the deeper flavors of artisanal bread, and lends depth and subtlety to pastries and other baked goods.
Some novice cooks and bakers are further confused to find references to "sweet" butter in their recipes. This is a slightly archaic term, used to differentiate between salted and unsalted butter. Since salted butter was by nature slightly savory, that made unsalted butter sweet in comparison. The term has no bearing on whether the butter is made with cultured or sweet cream, since sweet cream is the norm in the U.S.