Answer:
Turgidity is simply when a cell (usually plant cells as they have cell walls) is hypotonic with water compared to its environment, resulting in an enlargened and more firm cell, via the water pushing out against the cell wall. The cell wall is capable of resisting this water pressure, and hence explains why plant cells can only be turgid, as well as why plants can 'stand up' to gravity - because the firm turgidity supports the weight of the plant against gravity.
A turgid organism is an organism which is turgid, however I have not heard of a single case where animalia are turgid - as cell membranes cannot resist high water pressures within a cell (and if it did, it would just separate and pull the membrane apart) so, I believe it is only possible within organisms with a cell wall.