Vanishing of higher cohomology If $M$ is a manifold of dimension $n$, does singular cohomology $H^i(M, \mathbb{C})$ vanish when $i > n$ ?
If $M$ is an algebraic variety over $\mathbb{C}$, equipped with ordinary topology, can one say something about the vanishing of higer singular cohomology?
 A: If $M$ is a complex algebraic variety of dimension $n$, then its real dimension is $2n$, and so its singular cohomology vanishes in degrees $i > 2n$.  One can also try to say things about what happens in the range $0 \leq i \leq 2n$.
E.g. if $M$ is smooth, projective, and connected, then thought of as a real manifold, $M$ is compact, orientable, and connected of dimension $2n$, and so
$H^{2n}$ is one-dimensional.  More generally, Poincare duality relates $H^i$ and $H^{2n - i}$ (say with $\mathbb C$ coefficients).   
E.g. if $M$ is affine, then in fact $H^i$ vanishes if $i > n$.  This is a result of Mike Artin, which provides a reinterpretation of an earlier result of Lefschetz (what is commonly called weak Lefschetz).

In general, there is a lot of theory about singular cohomology of complex varieties.  In the smooth projective case, in addition to the information provided by real manifold theory (i.e. Poincare duality) one has additional input from Hodge theory.
In the general case, one has Deligne's mixed Hodge theory.
Again in the smooth projective case, one also has the hard Lefschetz theorem, which more or less can be interpreted as saying that the interesting cohomology appears in the middle degree, i.e. in $H^n$.
There is also the theory of nearby and vanishing cycles, which provides tools for describing how cohomology changes when a smooth variety degenerates under deformation to a singular variety.  (This is a complex analytic analogue of Morse theory.)

To conclude: this is a major area of investigation, and I have only described some of the basic tools and themes.  
A: I think the singular cohomology does vanish when $i>n$. You may prove it using long exact sequence or other standard tools. 
There is an in-depth discussion on your second question by Milne in first section of his notes. 
