What is the importance of the Poincaré conjecture? The Poincaré conjecture is listed as one of the Millennium Prize Problems and has received significant attention from the media a few years ago when Grigori Perelman presented a proof of this conjecture. But why is this interesting at all? What practical and interesting questions can be answered now that this conjecture is proven?
 A: The fundamental idea behind algebraic topology is to translate topological problems into algebraic problems. By so doing, one is able to strip unnecessary structure away from the study of manifolds, reducing it to a subject which can be tackled by group theoretical methods.
A typical problem in topology would be, given a group $G$, to classify the $n$-manifolds with fundamental group $G$. The easiest group for which one could ask this question is the trivial group, and the answer for $n=3$ is the Poincaré Conjecture. Poincaré's original vision would then have been, using this result and the techniques used to prove it as a stepping stone, to then classify $3$-manifolds with more interesting fundamental groups $G$. The idea would have been to mimic the classification of orientable closed surface by Euler characteristic. By so doing, $3$-manifold topology would be reduced to a branch of group theory. I expect that we will still see substantial work in this direction.
But an amazing thing happened. Instead of the Poincaré Conjecture only stripping away structure from $3$-manifold topology, its proof turned out to reveal a greater, richer structure on 3-manifolds. JSJ decompositions and geometrization mean that any $3$-manifold can be equipped with much more structure than at first meets the eye.
So, on the one hand, the Poincaré Conjecture gave a classification result- there are no exotic "pathological" homotopy spheres which fall through the coarse sieve of basic algebraic topology. On the other hand, it was a visible flag which lay in a domain which we didn't understand, and from the very beginning it was clear that the process of running to that flag and grabbing it would result in substantial gains for $3$-manifold topological understanding; this is indeed what happened.
A: While I'm sure an expert could give a much more informative answer, let me give a naive one.  In mathematics we are always interested in classification results.  If we want to understand an object that occurs in some problem and we have a classification result for that type of object, we can use it to gain traction on the problem.  For example we use the classification of semisimple Lie algebras or the classification of finite simple groups all the time.
The Poincare conjecture is part of a similar classification effort, but for closed 3-manifolds.  Now, closed 2-manifolds have a well-understood classification in a few senses; there is a topological classification, and there is also a geometric classification.  These classification results allow us to tackle many problems involving surfaces, such as the Riemann surfaces which occur when one analytically continues a holomorphic function.  So it is natural to look for a corresponding classification result in higher dimensions.  (In science journalism this is often justified as saying that topologists are trying to understand the shape of the universe, or something like that; you should feel free to take this justification or leave it.)
The topological classification of surfaces shows in particular that a surface is determined up to homeomorphism by its integral homology, so it's natural to ask whether the same is true for 3-manifolds.  Unfortunately, it's not; there is a famous example of a 3-manifold with the same homology as a sphere but which is not homeomorphic to it.  One way to prove this is to show that it has a nontrivial fundamental group, so now the natural question arises as to whether this is enough to uniquely identify the 3-sphere.  (If it is, then the problem of deciding whether a 3-manifold is the 3-sphere is in some sense purely algebraic, and in topology it is always desirable to reduce topological questions to algebraic ones since the latter tend to be easier.)
The problem became famous in part because it was extremely hard and in part because many mathematicians gave incorrect proofs.  This is often how problems become famous; if a problem in a field is hard that indicates that the tools of the field are not adequate to easily address it, so the problem spurs people to improve those tools.  
Perelman ended up proving the Poincare conjecture by proving a stronger result, the geometrization conjecture, which is the analogue for 3-manifolds of the uniformization theorem.  So work on the Poincare conjecture has led to a deeper understanding of 3-manifolds in general.  Not only that, but Perelman's work introduced important techniques which can now be used on other problems.  
A: Mathematicians are sometimes guided by what I call a “naïve formal” intuition. That is, they ask questions that are simple to state in a formal way, and sometimes get surprising answers. E.g.:


*

*I know how to multiply numbers. Can I multiply pairs? (Answer: ℂ makes ℝ×ℝ into an algebra.)

*Whole numbers decompose (multiplicatively) into "primes". Can I organise Lie stuff similarly? (Answer: ideals, semi-direct sum, semi-simple Lie groups)

*(a noncommutative geometry researcher shared a similar-sounding "outline of a programme" with me a few months ago, but I can't recall the details. They called it a "guiding philosophy" in only a slightly pejorative way—like "This isn't actually a result, and I want a concrete result, but this is what's guiding my search".)

*I can cone a plane to get a 2-sphere, cone a 2-ball to get a 3-sphere, and so on. Every time I do this, the number of rigid-rods coming out of points-in-space increases by 2 (one positive, one negative). This high-dimensional stuff isn’t so strange after all! Right? (Answer: the Milnor fibration shows that 7, 3, and 1 are special. This relates to why only 2, 4, and 8 can wear ℂ, ℍ,  structure.)


In the case of 2-manifolds, I guess you could make up a similar fake history: someone generalised flat surfaces to include deformed surfaces (hills, valleys, etc) using charts. Taking 2-manifolds and charts very literally led to ideas like "total curvature is related to genus" or "genus characterises compact 2-d surfaces".
But also, the “naïve” approach of “So what about the topology (connectedness) of higher-dimensional balls?” led to a flower-garden of ideas as mathematicians found they could treat some subset of cases using certain creative methods of argument. The flower-garden of new questions and approaches is what we really like, and is the reason the Clay Prize is worth much more than the million dollars of who gets the final reward for the "answer".
