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 : 

https://mathoverflow.net/questions/11226/commutative-rings-to-algebraic-spaces-in-one-jump/11234

**Erratum** : as nfdc23 points out, some condition on the diagonal is missing. The correct definition that I copy from Chris Schommer-Pries answer here 

https://mathoverflow.net/questions/16381/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

 1. $X$ is a sheaf on the big  étale topology on S,
 2. $\Delta : X \to X \times_S X$ is representable, and 
 3. 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 .