Since none of the others have addressed the types of surnames I'll add that to the mix. There are five styles of surnames:

*Given Name (which your last name falls under)

*Location Name

*Occupation Name

*nicknames/descriptors

*ornamental names

In the first category, given names, you have both your example where names like Williamson and Johnson derived from William's son or son of John, as well as clan names. In addition to patronymics (being named after your father) you also have matronymics (being named after your mother).

Location names came about in two ways. The first was with royalty. The name would link them with their place of origin and give an idea of their standing in society. The second path to derivation was with the rise of of respectably sized towns and cities. As you bring in more people you have to have ways to differentiate them. The easiest thing to note was if men originated from other communities. Daniel Radcliff, Jack London, and Edward Norton all have location based names.

The third category is occupational based names. Many common names you can think of fall under this category: Miller, Smith, Booker, Barber, Baker, Barker, Carter, Carver, Tanner, Hunter, and the list goes on.

The most obvious nickname derived surnames are color based names like Brown, Black, Gray, Green, Blue, and White usually referred to someones hair color, eye color, or complexion. Other names like Short came from physical descriptions not related to color. You could also have literal nicknames that over time became the family name. My last name falls into one of the later to subcategories. It's either a pet name for someone named Erasmus or a descriptive name for someone who looks like a fish.

So surnames primarily derived in larger cities as more people came together and you needed to differentiate between individuals. You needed to know the difference between Jack, The one with black hair, and Jack, the one from London. The custom spread as migration became more common and eventually certain names stuck. You went from being called John's son because your father was John to William's son because your father was also know as William's son even though you were the son of John. Sorry I don't have the books on hand to say whether they would have introduced themselves that way but it was clear that men would have been known that way.

Sources:

*Life in a Medieval Village by Frances and Joseph Gies

*The Waning of the Middle Ages by Johan

*Behind the Name for name examples

You have what is known as an "s" patronym, which suggests that you had an ancestor named William - the "s" indicating the "son of William." Typically, the "s" patronym in Britain is found in either Wales or Cornwall. Both areas were slow to adopt last names, and so when they did, many people used the identifier that they had - "Oh, that's Robert, the son of William" (i.e. Robert Williams).

One generation's specific identifier then became "fossilized" and passed down to future generations, even though Robert's children could have been known as Roberts. Because Robert adopted Williams as his formal last name, his son John, was known as John Williams rather than John Roberts.

The most famous/common example of an "s" patronym is found in the construction for "the son of John", which with a Welsh accent comes out as "Jones." One does not normally find this in Cornwall.

First, there is not one family named Williams, but many Williams families. It is an English translation of a Welsh naming form. Your ancestor was "William's Reece" or something like that. "Williams" started as a patronymic, a name part indicating who your father is. In older Welsh practice, that would be Rhys ap Gwillim, "ap" being the "son of" word, and his sister would be Nesta merch Gwillim. Up into the Renaissance it was normal for a Welshman in court to identify himself as a string of genealogy: John ap Rhys ap Gwillim ap Rhys P Hwyl and so on.

But people with the same family name may have no relation at all. They may just come from the same ethnic group. They may have taken the name to change their identity or to sound more normal in a new country. Family names from the British Isles only became common in the Rennaissance and fixed in the 1600s. So there were tons of people who took their name from their job - Smith, Miller, Shepherd, Weaver - without having ever seen each other. There was at least one Aunseremaker, which meant a clockmaker. Anyone who lived by the village green might wind up John atte green, and from there John Green, as might atte poole, atte wood (but that also became Attewood), and so on.

Source: Ingraham, People's Names