What are the Benefits of Using Algebraic Spaces over Schemes? I have heard that algebraic spaces have better formal properties than schemes. What are these benefits? Also, is there a natural way to go straight from affine schemes to algebraic spaces bypassing the locally ringed space construction? 
 A: 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 priori, if you want to work in the etale topology anyway, why not 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 Laumon, Moret-Bailly Champs algébriques.
(Edit: For "a fortiori" reasons to study algebraic spaces, I'll just say read the other answers :)
A: One of them was answered in response to question 1558 on when quotients of schemes by free group actions exist. When the group is finite, they exist as algebraic spaces. But, there are examples where they do not exist as schemes. So, being closed under quotients by free finite group actions is certainly nice.
I know the second part of your question is explained in the first couple of sections of Champs algebriques. They define a space as a covariant functor from algebras to sets that satisfies descent. Then, an algebraic space is such a functor that has an etale cover by the functors associated to some affine schemes. I don't really remember the details here.
A: In addition to being closed under taking quotients by etale equivalence relations, algebraic spaces are also closed under taking quotients by arbitrary finite group actions (need not be free, so the induced relation need not be etale). I only realized this recently, but I think the following argument is correct.
Suppose G is a finite group acting on an algebraic space X. Then the stack quotient [X/G] is a Deligne-Mumford stack (it has an etale cover by X). By the Keel-Mori theorem, DM stacks have coarse spaces. Let [X/G]→Y be the coarse space of [X/G]. Then Y is an algebraic space and the map [X/G]→Y is universal for maps from [X/G] to algebraic spaces. This means that Y is a categorical quotient of X by G. In fact, I think Y might actually be a geometric quotient (or at least a good quotient) of X by G, but I haven't unraveled all the definitions yet. 
A: The better formal property is that algebraic spaces are closed under taking quotients of etale equivalence relations (in practice typically coming from group actions), while schemes are not. 
One can define then directly from affines as done in Toën A master course in algebraic stacks, Cours N° 2 (in French; this source gives also a nice impression of the difference/similarity of schemes and alg. spaces and shows how they are exactly designed to support more quotients)
