Surnames like Clark, Baker, Taylor, Smith etc. are occupational, so if your surname is Taylor then your ancestor was a tailor and that's how he got his surname.
Like Preach said, names like Thompson, Jefferson, Richardson come from the name of the father. Some names from from the name of the mother, like Madison (from Madeline), Emmett (from Emma) or Elliott (from Elias). My surname, Sanders, means "son of Alexander" as does Sanderson.
Mac in Scotland also means "son of" (MacDonald, Mackenzie etc.), as does Fitz in Ireland (Fitzpatrick, Fitzgerald). Fitzroy meant "son of the king", used for illegitimate children of royalty. A Irish surname like O'Sullivan or O'Brien come from the grandfather's name.
In Iberian countries, Rodrigues meant "son of Rodrigo", Hernández meant "son of Hernando", Fernández "son of Fernando" etc.
In Eastern Europe, vić or vich means son, and sometimes yevna, ovna or ichna is used for daughter. The name Milošević means "son of Miloš", Abramovic means "son of Abram".
Names like Brooke, Hill, Byfield are based on geographical locations, i.e. your family lived near a brooke or a hill or by a field.
Names like Little and Armstrong were nicknames based on physical appearance. Someone with strong arms might be named Armstrong, someone who was short (or very tall) would be given the name Little.
