Motivation for the Mapping Class Group Question: What is the motivation for studying the mapping class group?  In particular, what types of questions does it attempt to answer and what kind of invariant is it?  
Motivation for this Question: Recently I've seen a number of references to things which assume knowledge of the mapping class group.  I've attempted to page through Dan Margalit and Benson Farbs' book on the subject as well as check out the wikipedia page for it, but both sources give the motivation as just "what it is" as opposed to, bluntly, "why should I care about this."
 A: The mapping class group (in genus $g$) is the fundamental group of the moduli space of compact Riemann surfaces of genus $g$.  Indeed, the latter space
is the quotient of a contractible space (Teichmuller space) by the mapping class
group.  Therefore a lot of the geometry of this moduli space is encoded in the
mapping class group.  
This is explained in the first one or two pages of the first chapter of (version 5.0 of) Farb and Margalit's book.  
A: With the risk of being redundant, I'll attempt to give my answer, if for no other reason than to help myself.  Then again, the best people to answer this question are probably people who use but don't exclusively study mapping class groups.
The mapping class group is a group one associates to a surface, and it's true that it distinguishes two surfaces that are "visually different."  However, I don't think this is where the true usefulness of it lies.  For one thing, it is a group of homeomorphisms up to equivalence (isotopy), so studying the mapping class allows us to say that the homeomorphisms of two surfaces are different (or at least one of them has "more" or different relationships between them).  This is still not the "best" use for the mapping class group though.
As mentioned in Matt E's answer, algebraic geometry is frequently interested in moduli space, and the mapping class group is the fundamental group of moduli space.  Therefore it bridges between algebraic geometry and the study of surfaces.
But what it really boils down to is the mapping class group is a group of isotopy classes of homeomorphisms and will show up any time you want to discuss homeomorphisms of a surface and often when you want to discuss homeomorphisms of higher dimension and it's nice to have properties of such a group any time you want to talk about homeomorphisms.  One thing I'm interested in, for example, is relating homeomorphisms of a base space and total space of a covering space and the mapping class group gives a language for this.
I think the Farb and Margalit book does a great job of motivating and in fact, is what first interested me in the topic.
A: A couple of things spring to mind. One is that if you are interested in $3$-manifolds, then the study of the mapping class group is the most natural thing in the world. This is because any compact connected $3$-manifold has a Heegard decomposition as the union of two handlebodies glued along their boundary. Thus the homeomorphism type of the $3$-manifold is controlled by the way the boundaries of the two handlebodies are identified. I.e. the $3$-manifold depends on the automorphism $\phi\colon \Sigma_g\to\Sigma_g$ of the boundary $\Sigma_g=\partial H_g$ of a genus $g$ handlebody. In fact, the homeomorphism type only depends on the isotopy class of $\phi$. So every mapping class gives rise to a $3$-manifold!
The other motivation that comes to mind is that a mapping class group of an $n$-punctured disk is isomorphic to the pure braid group on $n$ strands, which maybe is more obviously of interest. (I say "a" mapping class group, because you have to specify what happens to the boundary of the punctured disk, giving rise to different versions of the mapping class group, one of which is the pure braid group.)
