We can generalize the notion of Lindelöf spaces to consider spaces whose open covers admit a $\kappa$-sized subcover, for any infinite cardinal $\kappa$.
I imagine that $\kappa$ compact spaces in general have been studied in the past, and a single math.SE answer would likely not do justice to this whole topic. Thus, let's restrict ourselves only to $\mathfrak{c}$-compact spaces. That is to say, spaces whose open covers admit a continuum-sized subcover.
Is there a nice characterization of $\mathfrak{c}$-compact spaces?