Saturation current in inductor - Definition?
Saturation current only applies to non-air cored inductors or transformers. Basically the core material (iron or ferrite) can no longer linearly increase the amount of magnetic flux with an incremental increase in current.

Inductance is defined as the amount of flux produced per amp so for twice the current you'd expect twice the flux but if this ratio starts to reduce (due to saturation) the inductance is perceived to reduce and this can cause (in some circuits) an increase in current which lowers the inductance even further leading to a rapid hard-saturation situation.

Well-known is the B-H curve and this is a good measure of where saturation occurs: -



This is a rather genralized picture - a lot of inductors are reasonably linear then have a marked flattening at the top and bottom of the curve.

wiki has a useful article about it.