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 p2. 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 Fp2, and as a consequence, all such curves are defined over a finite degree extension of Fp.
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 Fp have j-invariant in Fp instead of just Fp2. 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 Gamma0(p) in SL(2,R) acts on the complex upper half plane with a genus zero quotient. For general p, this normalizer contains Gamma0(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.