Why do we have so many dualities in string theory? Why do we have so many dualities in string theory? Is there a reason for that?
 A: That's actually a good question, except I don't think anybody has a good answer. The dualities suggest that there is an underlying structure (sometimes called M theory) behind all the dual theories, and those dualities are a manifestation of that structure. Though we don't have a complete formulation of that structure yet, just the existence of this structure results in some consistency conditions which are those dualities - relationships between naively different theories and calculations.
You can have the following rough analogy. Think about the dual theories as different asymptotic expansions of some integral (say around zero and around infinity). Those expansions obey relations that express the fact they come from the same object. The state of affairs now is that we have those asymptotic expansions, and we discovered they obey relations that identify them as expansion of the same object, but we don't really have that object pinned down yet.
Relatedly, look at this answer I gave to someone's grandmother.
A: The qualitative way to understand the reason for string dualities, like almost everything else in string theory, is the holographic principle. The first thing you learn from black hole decay is that every exact global symmetry must be a local symmetry, because if you have no gauge field, the symmetry would not be conserved in black hole formation and evaporation. A black hole forgets about everything except the gauge fields on its surface.
Black hole decay also tells you that you can't make charges small in gravity without making masses small. The reason is that if you make a black hole from charged objects, it must be able to decay to enough charged particles to be left with no charge by the time it vanishes. In order for this to be statistically possible, the polarizing field on the horizon must be able to preferentially create charged particles. In order for this to happen when the quantum of charge is small, the charged particles generically must be light.
Further, gravity requires either charge quantization or massless charged particles, because if you have the proton, say, with infinitesimally more charge than the electron, you can throw a proton into a black hole and wait for an anti-electron to come out. The result will be an infinitesimal charge black hole, and this would have to decay to a particle with infinitesimal charge, which would have to be light compared to its charge. So if there is no charge quantization, there are lots of nearly massless teensy charged particles in the theory.
The holographic principle states that if you have an extreme charged black hole which is thermodynamically trivial (is zero temperature), you can describe the objects near it by the oscillations of its surface. This tells you that you should be able to describe strings near a stack of D-branes by oscillations of the D-branes themselves, without mentioning the strings. This is the principle of AdS/CFT.
Likewise, if I have a limit where certain pointlike black holes become light, I should be able to describe the entire string spectrum in terms of their motion against each other. This is the principle of Matrix theory.
The string dualities can be summarized as follows: every extremal black hole gives a dual description of the spacetime around it, and a complete description. The different descriptions have to be self-consistent, they have to contain the same physics. This is a modern version of the 1960s principle called "nuclear democracy", but it should be better called "gravitational democracy". Gravitational democracy demands a huge number of dualities, between the fundamental description provided by different types of black holes. 
Since gravity all but requires charge quantization, it is also is ok with monopoles. String theory requires magnetic duals for all objects, because a black hole or gravitational instanton can carry magnetic charge. Another source of duality, closely related to the first, is that quantum objects can carry magnetic charge.
Electric-magnetic duality tells you that you can't make big charges without making objects whose charges are small. This means that whenever you try to take a limit where everything is strongly interacting and heavy, the string theory requires that other things become light. The principle of gravitational democracy tells you that you can then formulate the theory in terms of the light objects only.
Take a string theory and compactify it on a circle. As the circle gets small the Kaluza Klein field of the circle gets a bigger and bigger quantum of charge. This means that the magnetic monopole gets a smaller and smaller charge, and therefore the lightest ones are lighter and lighter. This means that in the limit of tiny radii, the low-energy expansion of the S-matrix theory should be in terms of what were the magnetic variables in the original expansion. This is one way to motivate T-duality. The miracle of T-duality in type II string theory is that the new degrees of freedom are just different strings, and the duality never went through a regime where the strings are strongly coupled (only specific excitations became strongly coupled, and new excitations took their place as the light things after the duality)
The other class of dualities arise when you make the string coupling big. Then the gauge fields become strongly coupled, which makes something magnetically dual becomes light. In the case of type IIB string theory, you get the same theory again, because of its self-duality properties. In the type IIA theory, the D0 branes become light, and when you formulate the resulting theory in terms of these, you get the famous matrix theory of 11 dimensional M-theory.
