There is a whole collection of algebraically defined K-theories. My understanding is that algebraic K-theory is a presheaf of spectra $K$ on $\textbf{Sch}/S$ such that the homotopy groups of $K(X)$ give the algebraic K-groups of the $S$-scheme $X$.

What we can do is give $\textbf{Sch}/S$ a grothendieck topology, and find a fibrant replacement of $K$ in the local stable model structure on presheaves of spectra on $\textbf{Sch}/S$. These give us different K-theories. For example, if we take the etale topology (and take $K/n$) we get etale K-theory (with mod $n$ coefficients).

What is this process of finding a fibrant replacement geometrically? What is happening, and what does etale K-theory, say, measure?