When Queen Victoria married, she kept the name of her House: Hanover, but her children took the name of her husband's house: Saxe-Coburg and Gotha. That was the royal family's house until 1917, when it was changed to Windsor so that it would have an English name rather than a German one, given that England was at war with Germany at the time. The name "Windsor" was chosen because it was the location of one of the royal family's castles. Queen Elizabeth II was therefore of the House of Windsor, and when she married, she remained House of Windsor and her descendants would be Windsors, which did not please her husband, Prince Phillip, whose last name is Mountbatten. Eventually, though, she changed it so while her descendants are still House of Windsor, some of the descendants would have the last name Mountbatten-Windsor. There is some inconsistency in how it is used though, with some choice involved. Sometimes Charles and Andrew have gone by Windsor as a last name, other times Mountbatten-Windsor, depending on the circumstances.

As regards medieval Europe, much would depend on the country and when in the thousand-year span you are looking. The other answer refers to a modern time when English and German people had family names. For most of the middle ages no one had family names. People might have surnames (which are different) based on the domain they came from: Catherine of Aragon, Isabelle of Bavaria, Berengaria of Navarre. Up to the 1790's the French royal family had no family name: whn necessary, any member was "de France."

That being said, you need to understand that under Roman Catholic and most secular law, married women kind of ceased to exist. This is a hard concept for us, but we live after a lot of reforms. Simply, a married couple became one and her legal existence was subsumed in his. She could own no property, not even the clothes on her back: everything of hers was actually his, including her wages and their children. Women had no rights to their children: the father could keep her from any contact on mere whim. She could not sue for divorce for any reason because she could not bring any suit in law: he had to do any suing, including divorcing her.

In such a world, a woman's family could be some protection, the more so the higher their rank. But naturally she was always subsumed into her husband's family. That's why it was important to marry princesses to kings, princes, or at least dukes, that her marriage would not be a mésalliance.