There are two cases:

Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?

for topological manifolds: this is a generalization of the Poincaré conjecture.
Original problem posed, $n = 3$ for topological manifolds.
Millennium Prize Problems.
Last to be proven, only the 4-differential manifold case missing as of 2013.
Even the truth for all $n > 4$ was proven in the 60's!
Why is low dimension harder than high dimension?? Surprise!
AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.
For dimension two, we know there are infinitely many: classification of closed surfaces
for differential manifolds:
Not true in general. First counter example is $n = 7$ . Surprise: what is special about the number 7!?
Counter examples are called exotic spheres.
Totally unpredictable count table:
Dimension Smooth types
1 1
2 1
3 1
4 ?
5 1
6 1
7 28
8 2
9 8
10 6
11 992
12 1
13 3
14 2
15 16256
16 2
17 16
18 16
19 523264
20 24
$n = 4$ is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??

Dimension	Smooth types
1	1
2	1
3	1
4	?
5	1
6	1
7	28
8	2
9	8
10	6
11	992
12	1
13	3
14	2
15	16256
16	2
17	16
18	16
19	523264
20	24

Classification of closed surfaces

 0  0 

So simple!! You can either:

cut two holes and glue a handle. This is easy to visualize as it can be embedded in $R^{3}$ : you just get a Torus, then a double torus, and so on
cut a single hole and glue a Möbius strip in it. Keep in mind that this is possible because the Möbius strip has a single boundary just like the hole you just cut. This leads to another infinite family that starts with:
- 1: real projective plane
- 2: Klein bottle

A handle cancels out a Möbius strip, so adding one of each does not lead to a new object.

You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!

Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.