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 $p^2$. 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 $\mathbb{F}_{p^2}$, and as a consequence, all such curves are defined over a finite degree extension of $\mathbb{F}_p$.
There are two notions of supersingular prime: one is relative to a fixed elliptic curve over $\mathbb{Q}$, and one is absolute. For any elliptic curve $E/\mathbb{Q}$, 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 $\mathbb{F}_p$ have $j$-invariant in $\mathbb{F}_p$ instead of just $\mathbb{F}_{p^2}$. 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 $\Gamma_0(p) \in SL_2(\mathbb{R})$ acts on the complex upper half plane with a genus zero quotient. For general $p$, this normalizer contains $\Gamma_0(p)$ as an index 2 subgroup, with the nontrivial coset called the "Fricke involution" (a special case of Atkin-Lehner involultion). There is a standard order 2 representative, taking $z \mapsto -1/pz$. 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.
Edit: Thanks to Emerton for pointing out the connection. I'll try to expand on it a bit. The moduli problem of generalized elliptic curves with $\Gamma_0(p)$-structure has a coarse moduli space that is a smooth irreducible curve away from p, but has mod p fiber given by taking a disjoint union of two copies of $X(1)$ (a genus zero curve) and gluing along supersingular points (this description is more or less in Katz-Mazur, chapter 13). A geometric point describing an elliptic curve with j-invariant $\alpha$ is glued to the geometric point on the other irreducible component describing an elliptic curve with j-invariant $\alpha^p$. The Fricke involution switches the components, so the quotient of $X_0(p)$ by this involution is a genus zero curve glued to itself at supersingular points. The quotient has arithmetic genus zero if and only if all supersingular geometric points are glued to themselves - otherwise, the flat modular deformation to characteristic zero yields a smooth curve of higher genus. In other words, it is necessary and sufficient that all supersingular geometric points have no Frobenius conjugates, i.e., that the j-invariants of the supersingular curves lie in $\mathbb{F}_p$.