Singularities  
We can get clues about quantum gravity by thinking about
places where classical general relativity breaks down.
Such a place is called a singularity.
General relativity breaks down when there is some discontinuity in spacetime. Two important examples are
But there are other simpler kinds of singularities that string theory does know how to describe! The Big Bang and black hole singularities are of the spacelike type. This means that they extend throughout space and occur at a single point in time.
(It is obvious that the Big Bang
is a point in time. But why is the center of a black hole
spacelike? This seems counter intuitive but it is true!
The singularity of a black hole
is the end of time for an observer that falls into the black hole.)
The singularities that string theory can describe at the moment
are certain simple timelike singularities.
Such singularities exist at all times and are localized in space.
It turns out that there is a surprising connection between gauge theories, such as those that appear in particle theory, and the string theoretic description of such singularities. This fact is very important in various models that attempt to reproduce the standard model of particle theory. 

Geometry of extra dimensions  
String theory models predict ten spacetime dimensions
(nine space dimensions and one time).
Mtheory, which is a certain stronglycoupled
limit of string theory, predicts eleven spacetime dimensions.
The
extra dimensions
(six in string theory and seven in Mtheory)
are believed to be very small. The geometrical shape of
these dimensions and the singularities
in these extra dimensions can
explain the various types of particles in nature.
Although many important details, such as supersymmetry breaking, are unknown, the study of the possible shapes of the extra dimensions is important. This research field involves tools from advanced geometry and group theory. 

Field theories in higher dimensions  
String theory predicts the existence of consistent quantum field theories in spacetime dimensions higher than four. In particular there are very mysterious theories in six dimensions. Those theories are unlike any ordinary known quantum field theory. They do not have a coupling constant, and they have stringlike objects that are simultaneously electrically and magnetically charged. A better understanding of such theories will shed new light on nonperturbative effects (i.e., effects that cannot be Taylor expanded in the coupling constant and are therefore very hard to analyze) in four dimensional gauge theories. 

Nonlocal field theories and noncommutative geometry  
The requirement of local interactions is usually assumed
as a principle of quantum field theory.
It is, however, possible to drop this requirement and, at least in
theory, define a nonlocal quantum field theory.
It turns out that such nonlocal quantum field theories naturally arise in theoretical string theory. In such theories the nonlocal interactions are explained by fundamental strings that can stretch between different points of spacetime. These ideas are also related to the notion of noncommutative geometry. This is an abstract modification of analytic geometry in which coordinates are no longer ordinary numbers. Instead, they are taken to be abstract variables that do not commute. In noncommutative geometry the rules are very strange. For example, multiplying x by y is not the same as multiplying y by x. The order is important! The existence of such theories changes our fundamental notion of spacetime. It is also related to another important concept of particle and string theory: Supersymmetry. 
