Old money families in the U.S. are often WASPs. They have a very distinctive style that includes brands such as Lilly Pulitzer and Eliza B. Their children go to generally East Coast prep schools such as St. Pauls, Exeter, Choate, Andover and so forth. They summer in places like Bar Harbor, Nantucket, South Hampton, the Vineyard, etc.
In Europe, old money is synonymous with aristocracy. Dukes, counts, marquis etc. and their relatives. In France, Spain, and Italy they are typically Catholic and own hunting estates, and summer and ski in the same places. Their children go to Catholic schools, which can be in some cases single-sex.
P.S: Another person mentioned Paris Hilton as old money. This is not so as Paris' grandfather was the one to become rich, which means they are still very much new money. In fact, her attitude and attire probably epitomize new money.
The traditional American preppy style is much flashier than the old money European style, as all Americans are new money compared to European nobility.