Vgs is the gate to source voltage.
In the datasheet you'll find an absolute term Vgss this is the maximum voltage that can be applied between the gate and the source. Beyond this, you risk damaging the mosfet.
An N channel mosfet is essentially a P type sandwiched between two N type regions.
You are hosting a party and inviting all the neighborhood electrons to attend. So you broadcast "PARTY AT MA HOUZE YOLO #SWAG!". I.e. you apply a positive voltage at the gate with respect to the source. Given that your neighbors are some distance away (next door), your broadcast isn't loud enough (You are below Vgs(th)) Once you yell your invite loud enough (i.e. Have Vgs at Vgs(th)) your next door neighbors hear and come party with you.
Vgs(th) is the voltage at which the mosfet channel begins to conduct. At this voltage, a positive voltage, it creates an electric field, which attract electrons (since our applied voltage is positive, so positive charges on gate). These accumulated electrons near the gate, form a bridge between the source and the drain (which are both n type). Now you have a "continuous" n type path from source to drain.
You only attracted your next door neighbors, so your party is kinda lame. How do you get more people? You broadcast louder (Increase the range of your electric field - increase your Vgs).
Now, there's loud, and then there's HOLY CRAP, A PLANE IS ABOUT TO CRASH INTO OUR HOUSE, WHAT THE $#@! IS THAT NOISE!?! We don't want to freak people out too bad, so we need to know how much louder (the difference between) our call is than the minimum needed for just our immediate neighbors (Vgs(th)). This difference is called by a couple of different names, but the two that I've heard most often are V-on (Von) or V-overdrive (Vov). This quantity (Vgs-Vth) represents how much more potential is between the gate and the source than is needed for the transistor to turn on, and it influences just about every other behavior of the MOSFET: current in triode (how many people are at your party when just your neighbors can hear), current in saturation (how many people are there when it's full of people), and transconductance (how much volume you need per person at the party), just to name a few.
So now you've increased your Vgs to the point that you cannot accept anymore people on your property. You are completely full. You can broadcast your party as much as you want, but there simply isn't enough room to accommodate everyone. Your transistor is now in saturation. [Technically, more people can be at the party: if your immediate neighbors start broadcasting the party (increasing Vds), and people start partying at their houses (the source and the drain), this increases the number of people at the party (increased current). This is called channel-length modulation. However, this can only happen if your house is already full (channel-length modulation only happens when the device is in saturation).]
If you start increasing your Vgs to the point of Vgss, well...cops show up and shut you down. Underage drinking, drug use etc.. You go to jail (Your transistor has been damaged).