The essence of the Poincare conjecture: he conjectured that every simply connected 3D-manifold is homeomorphic to the sphere.
In non-maths language, Poincaré's conjecture is that if you have a shape which everywhere looks like 3D-space (much like our universe seems to) [this is what makes it a "3D-manifold"], and this space is "finite" in some sense and without boundary [this is the "closed" part, for manifolds] in a way such that whenever you draw a loop in your space you can pull it to a point (like you can pull a rubber band on a regular sphere to a point, no matter how you wrap it, whereas there are non-trivial loops on the surface of a doughnut, or a "torus) then this space, up to a bit of deformation, must be a 3D-sphere. The language is a little misleading here - "simplest shape in 3 dimensions" isn't really what is meant. I can immerse a normal sphere "in" 3 dimensions, what is important is the internal dimension of the sphere, as described above. A 3D-sphere isn't the surface of a normal ball that you'd be used to, it is its analogue which can be immersed in 4D-space, but if you lived on one it would "feel" like you lived in 3D-space. You couldn't draw it "in" 3D-space.
The reason for the 3D-sphere, and not n-sphere for any n, isn't that the same statement doesn't work for any n, it's just that historically the 3D case was the hardest to solve. It's an interesting phenomenon in topology that some things become easier to prove in high enough dimensions (in some sense, you get more room to work with - there is a standard trick called the Whitney trick which works in dimensions 5 and above which is often the reason for some things in topology being solved in high enough dimensions). Of course, it's not as simple as saying "everything gets simpler". It's possible to show, for example, that an effective classification of manifolds of dimension 4 and above is impossible.
Probably the most famous event in mathematics in the last 25 years was Wiles' proof of Fermat's Last Theorem, for which Wiles received numerous honours. But the complete proof of Fermat's last theorem depends on a result proposed by J. P. Serre and proved by Ken Ribet, a conjecture by Taniyama and Shimura suggested the path to follow, too many mathematicians to list here made other contributions, and finally Wiles' "proof" contained a mistake which was pointed out by Richard Taylor (who helped Wiles fix the mistake).
People read about celebrities lives, what they do on a day to day basis and care about just everything they do. This is entirely the opposite in science, with a few notable exceptions. In science Fermat's last theorem is famous, Andrew Wiles is not. The law of cosines is famous, Francois Viete is not. Electromagnetism is famous, Heaviside is not. Gauge invariance is famous, Hermann Weyl is not. It is with very few exceptions the theories and discoveries that are remembered, not the names. The only exceptions I can think of are Einstein, Newton, Hawking and maybe Aristotle.
Hamilton was kind of frustrated and jealous of the fact that Perelman solved Poincaré using Ricci Flow as the basis, but he could not do it for almost 2 decades even after pioneering Ricci Flow. There is a book "Perfect Rigor" by Gessen (It's rather unfortunate Gessen does not attempt a more balanced biography; read it for the facts not for Gessen's opinions). I would suggest all math aficionados to read it to understand the man who solved Poincaré. There are inevitably arbitrary variables in life that make it "unfair." Richard Hamilton fell victim to one of these "unfair" circumstances: I think he was "too old" when Grigory Perelman was given the Fields Medal recognition, a recognition that Hamilton, too, should have been included, due to the former's foundation work upon which Perleman used to accomplish his own work. The arbitrary cut-off age that qualifies a person for the Fields Medal is 40. If you turn 41 a few hours or days or weeks before the announcement, you don't qualify, regardless of the quality of your mathematical contribution. But, as the philosophy of the medal puts it: the Fields Medal is to "encourage" the younger and promising mathematicians, NOT to "reward" some ultimate or crowning accomplishment in mathematics. Once you accept such an arbitrary rule, it is an arbitrary rule that is easy to live with, however you feel toward it personally. If the Riemann Hypothesis was solved tomorrow, it would be known about whether or not it was a known name or someone from obscurity that solved it. And either way, no one would care about the person, no one would want to know how they live their life, what events they go to, or what they enjoy.
Wow, I'm really amazed at how many false statements people make about this kind of stuff, and if you're new to this field then you're likely to be very confused by learning incorrect things. First of all, this area of mathematics is not topology; it's differential geometry. That is like saying that calculus is in the field of arithmetic. Also, the normal sphere we think about is not a "3-dimensional sphere," but a 2-dimensional sphere sitting in 3-dimensional space. Calling that object a "3-dimensional sphere" is just confusing because then people get the idea that you're talking about the entire ball, so including the inside of it. You are only talking about the surface of that ball, which is a 2-dimensional surface, hence it is a 2D-sphere. A 3D-sphere is the surface of a 4-dimensional ball, and these concepts are important when you want to abstract these objects from the Euclidean spaces they are living in and consider the differential structure of them without a reference space."
Morgan and Tian's is still one of the best treatments out there when it comes to understanding Ricci Flow.