The GFCI measures the current through the hot wire and compares it to the current in the return wire. If they are the same, all is good. If not, the current is flowing through the chassis of the unit or through your body or some other undesireable path. When this happens, the GFCI opens an internal switch so there is no more power to the plug.
It should not affect anything if you plug one or ten GFCIs into each other.