A topological space $X$ is called quasi-compact if every open covering of $X$ has a finite subcovering.
(This is the terminology adopted by algebraic geometers to emphasize that Hausdorffness is not required. Compact spaces are then Hausdorff quasi-compact spaces).
A topological space $X$ is said to be noetherian if every non-empty family of closed subspaces has a minimal element.
For example $\mathbb A^n_k$ and $\mathbb P^n_k$ are noetherian: this follows from Hilbert's theorem that the polynomial ring $k[T_i,...,T_n]$ is noetherian .
The following are then equivalent for a topological space $X$:
$\bullet $ $X$ is noetherian.
$\bullet $ Every subset of $X$ is quasi-compact.
Since projective space $\mathbb P^n_k$ over a field is noetherian , any quasi-projective variety is quasi-compact because by definition it is the intersection of an open and a closed subset of some $\mathbb P^n_k$, and any subset of $\mathbb P^n_k$ is quasi-compact by noetherianity of $\mathbb P^n_k$.
Every quasi-projective variety is quasi-compact.
Edit: as emphasized by Pete, even an arbitrary subset of a quasi-projective variety is quasi-compact, since it too is a subset of $\mathbb P^n_k$.