If someone (in the world or near you) has the same last name as you do does that make them related to you?

[* Xanthippe]

You have such great answers already (particularly Ashley's - I doubt I could have explained it half as clearly), there is little left to say. I would just add that the reason Smith is the most common surname in the British Isles is that pretty much every village had one - a blacksmith that is. So you are no more likely to be related to another Smith as you are to someone with a completely different surname.