What is a graviton?

A graviton is a theorised particle (specifically a boson). It is said to be its own antiparticle.

It has a special role in general relativity (being that the science is all about gravitation and the warping of space/time).

In relation to your mention of string theory, it is said to be a closed string with a very low-frequency vibration. Closed strings are loops which vibrate at different energy levels like the one below.

Vibrating closed string
(source: superstringtheory.com)

In quantum field theories, interactions are exerted by the exchange of a force carrier particle. For the electromagnetic force, this is the photon, for the strong force, the gluon and for the weak force, the W and Z bosons. All these force carrying particles have been observed.

Now it is imagined, that if we are able to find a quantized version of gravity, it would also be described by a quantum gauge field theory. Then the force would be exchanged by a messenger particle as well. This messenger particle is the graviton. It has spin 2. We haven't observed this particle yet.

A graviton is a particle that carries gravitation.

So as to explain that: imagine you have two point particles moving away from each other. Particle A sends a graviton towards particle B, and starts moving towards particle B. Then particle B receives the graviton and starts moving towards particle A.