Here is one intuitive way to think about it:
A scheme is something which is Zariski-locally affine, whereas an algebraic space is something which is etale-locally affine.
One way to make this precise: a scheme is the coequalizer of a Zariski-open equivalence relation, whereas an algebraic space is the coequalizer of an etale equivalence relation.
Another way is to say that, as a functor on rings, a scheme has a Zariski-open covering by affine functors, whereas an algebraic space has an etale covering by affine functors (thus bypassing reference to locally ringed spaces, regarding your second question).
Why do we care? A fortioripriori, if you know you're just going want to end up working work in the etale topology anyway, why not start over and fix the definition of scheme to say "etale-locally affine" instead of "Zariski-locally affine". This is just one motivation for studying algebraic spaces, which you can read more about in Champs algébriques.
(Edit: For "a fortiori" reasons to study algebraic spaces, I'll just say read the other answers :)
