Type II superstring theory

The second and third flavor of superstring that appears in 10-dimensions are the type IIA and type IIB superstring theories. 

Both of these theories have the maximum amount of supersymmetry: 32 supercharges in 10 dimensions.

Both the Type IIA and Type IIB superstring theories are based on oriented closed strings. 

On the worldsheet (history traced out by string), they differ only in the choice of GSO projection, which is am ingredient used in constructing a consistent model of superstring theory. 

Type IIA superstring theory

Type IIA superstring theory, at low energies is described by type IIA supergravity in 10 dimensions. This is a non-chiral theory or it is left-right symmetric. It has (1,1) d=10 supersymmetry. That anomalies cancel in this theory is trivial. 

Edward Witten, building on the work of Michael Duff and Paul Townsend, realized that the limit of type IIA superstring theory where the string coupling goes to infinity, becomes a new 11-dimensional theory known as M-theory. 

The mathematics of type IIA superstring theory, primarily is found in symplectic topology and algebraic geometry. This is particularly Gromov-Witten invariants.  

Type IIB superstring theory

Type IIB superstring theory, at low energies, is described by type IIB supergravity in 10 dimensions, which is a chiral theory or is left-right asymmetric. It has (2,0) d=10 supersymmetry. That anomalies cancel in this theory is not trivial. 

In the 1990s, it was realized that the type IIB superstring theory with coupling constant "g" is equivalent to the same theory with coupling constant "1/g." This equivalence is known as S-duality. 

Orientifolding type IIB string theory gets you type I superstring theory.

Type IIB's mathematical treatment belongs to algebraic geometry and the deformation of complex structures which was originally studied by Kunihiko Kodaira and Doland C. Spencer. 

In 1997, Juan Maldacena proposed that the N=4 supersymmetric Yang-Mills theory in the 't Hooft limit is equivalent to type IIB superstring theory. This was the first real consideration of the AdS/CFT correspondence.  

T-duality

It was realized in the late 1980s that type IIA and type IIB superstring theory are related to each other by T-duality.