If I remember correctly, this goes roughly as follows. Consider the category $\mathcal C=\operatorname{Rings}^{op}$, first endowed with the Zariski topology. You can consider sheaves on this site that are locally covered by representable sheaves. Such sheaves form a category equivalent to the category of schemes.
As you can guess, if you now consider $\mathcal C$ endowed with the étale topology, you will get a category equivalent to the category of algebraic spaces.
ps : I am looking for a reference. The best I found by now :
Commutative rings to algebraic spaces in one jump?
Erratum : as nfdc23 points out, some condition on the diagonal is missing. The correct definition that I copy from Chris Schommer-Pries answer here
Quasi-separatedness for Algebraic Spaces
is the following
Definition: An algebraic space over $S$ is a functor $X : (Sch/S)^{op} \to S_{et}$ such that
- $X$ is a sheaf on the big étale topology on S,
- $\Delta : X \to X \times_S X$ is representable, and
- there exists an $S$-scheme $U \to S$ and a surjective étale morphism $U \to X$.
This is Definition 5.1.10 in Olsson's book Algebraic Spaces and Stacks https://bookstore.ams.org/coll-62/ . In remark 5.1.11 he remarks that Knutson's definition includes the fact that $\Delta$ is quasi-compact.
The same definition and more information can be found in the stacks project : see https://stacks.math.columbia.edu/tag/025Y and https://stacks.math.columbia.edu/tag/076M .