This is a theorem of Deuring, 1941. David alluded to this, but Section 5.3 of Silverman's The Arithmetic of Elliptic Curves has a proof that 5 conditions concerning elliptic curves over a characteristic p perfect field are equivalent, and any one of them can be taken as the definition. There are additional definitions of supersingularity, concerning the vanishing of the Hasse invariant (a modular form mod p defined by the eigenvalue of Frobenius acting on the Serre dual to the invariant differential), or line bundles with trivial p-th tensor power being automatically trivial. My favorite definition is that the kernel of multiplication by p is a connected group scheme (necessarily of order p^2).
By analyzing the l-adic Tate module, you can conclude that the endomorphism ring of any elliptic curve is a free Z-module of rank 1, 2, or 4, and for rank 2 (resp. 4), analysis of dual isogenies shows that the ring has to be an order in an imaginary quadratic field (resp. a quaternion algebra). If you assume a supersingular curve has endomorphism rank 1 or 2, you can derive a contradiction by combining two facts:
- There are only finitely many isomorphism classes of supersingular curves for a given prime p (in fact, the total number weighted by automorphisms is (p-1)/24). This uses the fact that the j-invariant of a supersingular curve lies in a finite field.
- Given a curve whose endomorphism ring has rank 1 or 2, any isogenous curve has an endomorphism ring with the same fraction field.
Take a sequence of elliptic curves as successive images of cyclic l-isogenies, where l is chosen to be prime in the ring of integers of the fraction field. Two of them will be isomorphic, so you get an endomorphism by a cyclic isogeny. Analysis of degree shows that this endomorphism is equal to an automorphism composed with multiplication by a power of l, and the two endomorphisms have nonisomorphic kernels.