Is there an intuitive way of thinking about the extra dimensions in M-Theory? Why are 11 dimensions needed in M-Theory? The four I know (three spatial ones plus time) have an intuitive meaning in everyday life. How can I think of the other seven? What is their nature (spatial, temporal, ...) and is there an intuitive picture of what they are needed for?
 A: This should be a comment since I am not a string theorist but its too big. When Luboš (Luboš correct me if I'm wrong) speaks of the "shape" in his comment:

They're spatial dimensions - new temporal dimensions always lead to at least some problems if not inconsistencies - but otherwise they're the same kind of dimensions as the known ones, just with a different shape. To a creature much smaller than their shapes' size, they're exactly the same as the dimensions we know. The theory implies that the total number of spacetime dimensions is 10 or 11. We don't have "intuition" for higher-dimensional shapes because the extra dimensions are much smaller than the known ones but otherwise they'er intuitively exactly the same as the known spatial dimensions

he means that the higher dimensions are "compactified". A simple example of a compact space is a circle, or a Cartesian product of circles (a torus) or a high dimensional sphere. The crucial idea is that they are topologically compact, meaning roughly that they are finite and closed i.e. they have no boundary just like the torus or sphere have no boundary. So Luboš's little creatures would return to their beginning point if they walked far enough in the same direction.
As I understand it, one of the proposed "shapes" for the compactified dimensions is the Calabi-Yau manifold. Wholly for gazing on beauty's sake, its worth also looking these here at the Wolfram demonstrations site. Be aware that you're looking at a projection, hence the seeming "edges" are not the manifold's boundary. Like the torus and the sphere, these manifolds would let a little creature return to their beginning point eventually by travelling in a constant direction and nowhere would they come across a barrier or boundary.
Actually, it's not out of the question that the three spatial dimensions of our wonted experience are like this too, just that we're talking awfully big distances (10s to 100s of billions of light years) for us to come back to our beginning points if we blasted off into space and kept going in the same direction. As I understand this, this idea is seeming less and less likely since our universe globally is observed to be very flat indeed. See an interesting discussion at MathOverflow on what the fundamental group of the Universe might be like
Update: See also this answer clarifying some of my description of compactified dimensions. If they are big enough (as for our everyday three spatial dimensions, if they are compactified too) even though a constant direction vector can be integrated to a closed loop through space, the fact that the Universe is expanding means that one cannot traverse this loop in a finite time. 
A: Ok, this question requires a more careful answer than what was presented here. First, extra-dimensions appear in string theories or M-theory (which is in fact not a well defined or well known theory, if any). Considering only the bosonic string we have the Weyl invariance. If you calculate the energy momentum tensor then the Weyl invariance implies that its trace must vanish. This doesn't happen in general and a path integral quantization of a bosonic string will show that you can restore weyl invariance only if the dimension of the space you are working with is 26. If one adds fermios you will get a somehow different relation but the main idea is the same: in order to restore conformal symmetry you must work on a fixed number of dimensions, specifically 10. Of course there are several issues with this way of thinking. The first and most important one: string theory is not a theory of nature, it is just an dimensional extension of a perturbative expansion without an underlying theory per se so there is no good natural reason for string theory to tell what properties nature should have. If string theory has anomalies then this is the problem of string theory and not of nature. Second, one should in principle be able to construct theories that are "sub-critical" i.e. they do not have to obey the dimension rule. This has been done with modest success. Now, compactification is a process of imposing some sort of compactness for the supplemental dimensions appearing in critical string theories. 
