Separated and Finite Type Scheme over an Algebraically Closed Field

As you note, finite type means that $X$ is the union of finitely opens each of which is the Spec of a finite type $k$-algebra. This is an abstraction of a basic finiteness property of quasi-projective varieties.

Separatedness is the analogue, for the Zariski topology, of being Hausdorff, and like Hausdorfness, it often plays a basic role in arguments. As one example, if $f:X \to Y$ is a morphism of $k$-schemes and $Y$ is separated, then the graph $\Gamma_f \hookrightarrow X \times Y$ will be a closed subscheme.