A generalized inequality is a partial ordering on $\real^n$ that has many, but not all, of the properties of conventional inequalities and the standard ordering on $\real$.
Formally, a generalized inequality is defined with respect to a proper cone $K$ in the following manner:
\[\x \gle \y \Longleftrightarrow \y - \x \in K.\]The cone in question is usually clear from context, but if not it can be specified with the notation $\gle_K$.
For example, with vectors, unless otherwise specified the cone is assumed to be $\real_+^d$, the nonnegative orthant. Thus, $\x \gle \a$ means $\a - \x \in \real_+^d$, or $a_j - x_j \ge 0$ for all $j$.
An example where this idea is relevant to statistics is the CDF of a multivariate distribution, where $F_\x(\a)$ means $\Pr{\x \gle \a }$.
The strict inequality is defined similarly, with $\x \gl \a$ defined to mean $\a - \x \in \textrm{ interior} (\real_+^d)$, or $x_j < a_j$ for all $j$.
For matrices, the cone is usually taken to be the set of positive semidefinite matrices, such that $\A \gle \B$ means that the matrix $\B - \A$ is positive semidefinite (and $\A \gl \B$ meaning that $\B - \A$ is positive definite).