When is "Next Saturday"?

"The following Saturday" is the phrase that best describes the second coming Saturday (i.e., the second Saturday from today). "Next Saturday" is short for "the next Saturday" which means exactly what it sounds like... the very next Saturday which is coming up. "Next" is what causes all the problems, of course, because one can argue that it means not this Saturday coming but the next one after that. It is such an equivocal term that it really shouldn't be used at all. It's better if one intends to specify the upcoming Saturday, he/she just says "this Saturday".

Edit: Next week means not this week, but the next. In the same vein as what I mentioned above, it means the very next week that is coming up. Just like if it was Saturday and you were to say "next Saturday", we are in this week, and next week will be... well, next.

Next means whatever the speaker intends it to mean. Some listeners think it is the first coming like in “next exit or first in line”, while others think it means the day in the following week like “next Saturday next week”.

Trust but verify your assumptions, I say. Don’t assume wrong and jump to conclusion! Anything can mean anything under agreement.

Next Saturday typically means the next Saturday that occurs, within two to seven days. It wouldn't usually be within one day, because if it's Friday, you would usually say "tomorrow," instead. If you're talking about a Saturday that is more than seven days away (with another Saturday in between), you would usually say "a week from Saturday" or "two weeks from Saturday," up to about three weeks. Beyond three weeks away, you would probably give a specific calendar date: Saturday, May 19.

Of course, people get confused about time when they're speaking, and it never hurts to verify.

"Next week" doesn't mean the same thing at all. Because the standard work week is Monday through Friday for most people, when they say "next week," they usually mean in the next Monday to Friday time range after the current one. People who don't work standard work weeks just adapt their understanding to that, too. If people mean a Saturday or Sunday "next week," they'll usually specify. "Saturday, next week" might mean the same thing as "a week from Saturday" in some contexts, but it's more confusing.

It usually is the Saturday that falls on the coming week.

"I'll see you next Saturday"

This means that the speaker intends to meet the listener on the Saturday, in the next week. If he had wanted to convey to the listener that he wanted to meet the latter on the Saturday of the same week, he would've said:

"I'll see you on Saturday"

or

"I'll see you this Saturday"

Although the first sentence could invoke a question "Which Saturday?", the speaker could clarify that he wanted to meet him on the coming Saturday.