If string theory is a theory of gravity, then how does it compare with Einstein's theory of gravity? What is the relationship between strings and spacetime geometry?

Strings and gravitons

    The simplest case to imagine is a single string traveling in a flat spacetime in d dimensions, meaning that it is traveling across space while time is ticking, so to speak. A string is a one-dimensional object, meaning that if you want to travel along a string, you can only go forwards or backwards in the direction of the string, there is no sideways or up and down on a string. The string can move sideways or up and down in spacetime, though, and as the string moves around in spacetime, it sweeps out a surface in spacetime called the string worldsheet, a two-dimensional surface with one dimension of space and one dimension of time.
    The string worldsheet is the key to all the physics of the string. A string oscillates as it travels through the d-dimensional spacetime. Those oscillations can be viewed from the two-dimensional string worldsheet point of view as oscillations in a two-dimensional quantum gravity theory. In order to make those quantized oscillations consistent with quantum mechanics and special relativity, the number of spacetime dimensions has to be restricted to 26 in the case of a theory with only forces (bosons), and 10 dimensions if there are both forces and matter (bosons and fermions) in the particle spectrum of the theory.
    So where does gravity come in?
    If the string traveling through spacetime is a closed string, then the spectrum of oscillations includes a particle with 2 units of spin and zero mass, with the right type of interactions to be the graviton, the particle that is the carrier of the gravitational force.
    Where there are gravitons, then there must be gravity. Where is the gravity in string theory?

Strings and spacetime geometry

    The classical theory of spacetime geometry that we call gravity consists of the Einstein equation, which relates the curvature of spacetime to the distribution of matter and energy in spacetime. But how do the Einstein equations come out of string theory?
    If a closed string is traveling in a curved spacetime, then the coordinates of the string in spacetime feel this curvature as the string propagates. Once again, the answer lies on the string worldsheet. In order for their to be a consistent quantum theory in this case, the curved space in which the string travels must be a solution to the Einstein equations.
    Now this is really something! This was a very convincing result for string theorists. Not only does string theory predict the graviton from flat spacetime physics alone, but string theory also predicts the Einstein equation will be obeyed by a curved spacetime in which strings propagate.

What about strings and black holes?

    Black holes are solutions to the Einstein equation, therefore string theories that contain gravity also predict the existence of black holes. But string theories give rise to more interesting symmetries and types of matter than are commonly assumed in ordinary Einstein relativity. So black holes are more interesting to study in the context of string theory, because there are more kinds to study.

Is spacetime fundamental?

    Note that there is a complication in the relationship between strings and spacetime. String theory does not predict that the Einstein equations are obeyed exactly. String theory adds an infinite series of corrections to the theory of gravity. Under normal circumstances, if we only look at distance scales much larger than a string, then these corrections are not measurable. But as the distance scale gets smaller, these corrections become larger until the Einstein equation no longer adequately describes the result.
    In fact, when these correction terms become large, there is no spacetime geometry that is guaranteed to describe the result. The equations for determining the spacetime geometry become impossible to solve except under very strict symmetry conditions, such as unbroken supersymmetry, where the large correction terms can be made to vanish or cancel each other out.
     This is a hint that perhaps spacetime geometry is not something fundamental in string theory, but something that emerges in the theory at large distance scales or weak coupling. This is an idea with enormous philosophical implications.