There are many equivalent ways to define supersingularity for an elliptic curve over a characteristic p field.  One of them is that the p-torsion of the curve is connected, i.e., it is a purely infinitesimal group scheme of order p<sup>2</sup>.  As Jonah mentioned, supersingular means very special, and is not a statement about smoothness.  There is a theorem of Deuring that implies the j-invariant of a supersingular elliptic curve always lies in F<sub>p<sup>2</sup></sub>, and as a consequence, all such curves are defined over a finite degree extension of F<sub>p</sub>.  

There are two notions of supersingular prime: one is relative to a fixed elliptic curve over Q, and one is absolute.  For any elliptic curve E/Q, a prime p is supersingular for E if E has good supersingular reduction at p.  Such primes are known to be asymptotically density zero, but infinite in number (by a theorem of Elkies).  Lang has a conjecture regarding precise asymptotic behavior.

Supersingular primes in the absolute sense are those primes p for which all supersingular elliptic curves over an algebraic closure of F<sub>p</sub> have j-invariant in F<sub>p</sub> instead of just F<sub>p<sup>2</sup></sub>.  These happen to be the primes that divide the order of the monster simple group, and they are also the primes for which the normalizer of Gamma<sub>0</sub>(p) in SL(2,R) acts on the complex upper half plane with a genus zero quotient. For general p, this normalizer contains Gamma<sub>0</sub>(p) as an index 2 subgroup, with the nontrivial coset called the "Atkin-Lehner involution".  There is a standard order 2 representative, taking z to -1/pz.  The quotient curve classifies unordered pairs of elliptic curves with dual degree p isogenies between them.  I do not know any canonical relations between these characterizations of supersingularity.