A bilinear form
f(u,v) is called positive semidefinite
if for all vector
s v, f(v,v)>0.
Since a quadratic form is represented by some square matrix A, we can immediately "lower" the property to matrices: A is positive semidefinite iff for all vectors v, vtAv > 0.
Positive semidefiniteness extends some properties of "positiveness" of real numbers. For instance, the second derivative of a function f:Rn->R is an n*n matrix. If ∇2f(x) is positive semidefinite, then f is (locally) convex near x. In one variable, you already know this: if f:R->R satisfies f''(x)>0, then f is convex near x.