Actually most (except wet electrolytic) capacitors is solid-state
. But when it used in context of motherboards/power supplies it means only "Polymer electrolytic", to emphasize difference from liquid electrolytic. Polymer capacitors have lower ESR and can handle higher ripple current, and ESR not grows on high temperatures. See http://en.wikipedia.org/wiki/Polymer_capacitor
"Solid state capacitor" is marketing, not technical term.