if you want to look at your chart, the saturation region is one where the traces are going up: where Vds is very low and Ids goes up with Vds, given a Vgs. the left most region where you put the "on" state symbol.
the linear region is where Vds is significant and does NOT vary with Id. Instead, Id goes up and down with Vgs. That's the middle area where the traces are flat.
the "off" area is where Vgs is < Vgs(th).
Added after 4 minutes:
another way to look at the same thing is to look at its Id vs. Vgs chart: from irfp240.
the left most region is where Vgs<Vgs(th), so the device remains off (Id is very small).
as Vgs goes above Vgs(th), the mosfet starts to conduct and its Id goes up with Vgs. this is the linear region.
As Vgs continues to go up, Id stops increasing as the mosfet is "saturated".
the same thing holds for bjt as well.
Added after 34 seconds:
and you can find the same relationship in ALL mosfet datasheets.