Higher Dimensional equivalent of genus For 2-manifold there exists the notion of genus.
I (as a non topologist) was wondering if there exists something similar for d-manifolds.
Thank you
 A: If you study an $n$-dimensional differential  manifold $M$ , the simplest and  most canonical  invariants attached to it are its De Rham cohomology $H^k_{DR}(M,\mathbb R)$, which are $\mathbb R$- vector spaces.
They are completely intrinsic since they are defined in terms of smooth global differentiial $k$-forms.
De Rham's celebrated theorem says that they coincide with the singular cohomology vector spaces defined topologically $H^k_{sing}(M,\mathbb R)$.
 The advantage of De Rham's point of view is that you need much less machinery to calculate them, witness Loring Tu's magnificent An Introduction to Manifolds 
If the manifold is compact, the cohomology vector spaces are finite-dimensional and  their dimensions $b_k=dim_\mathbb R H^k_{DR}(M,\mathbb R)$ are important numerical (=integer-valued) invariants, the Betti numbers of $M$,  vanishing for $k\geq n+1$.
 The alternating sum $\Sigma (-1)^kb_k$ is the Euler characteristic mentioned by Sasha.
However in higher dimensions I think it is better not to try to have just one number summing-up the properties of $M$, but to consider all the Betti numbers simultaneously.
And other invariants as well: for example the fundamental group $\pi_1(M,m_0)$ based at $m_0\in M$. It is purely topological but more difficult to compute.  
I have interpreted "manifold" as "differential manifold".
If you want to study topological manifolds you can replace De Rham cohomology by singular homology.
You will get finer invariants, but you will have to invest more time in the prerequisites .
A book I can recommend is  Lee's  Introduction to Topological Manifolds
