What does next Saturday mean?

“Next Saturday” means the very next Saturday, whether it’s in two days or nine days. “On Saturday” and “this Saturday” mean exactly the same thing. A Saturday that falls more than eight days away is “a week from Saturday.” – joiedevivre.

Does next weekend mean this weekend?

If the week has already started, “next weekend” would be the ending of the following week, “next” being key, referring to next week’s week end. To refer to the weekend of the week you are in you should say “this weekend”, “this” meaning this week we are currently in.