Type II superstring theory starts from the assumption that small perturbations of the vacuum state result only in orientable closed[*] strings.
By contrast, Type I superstring theory starts from the assumption that perturbations near the vacuum state can be either open or closed strings, but both must be non-orientable.
Another difference is that while Type II theories have two 10-dimensional supersymmetry generators, Type I theories have only one. This difference is a consequence of the non-orientability of Type I strings. Assuming the strings are non-orientable means forcing the positive and negative chiral components of the worldsheet spinors to be dependent on each other. They are both determined by the same set of modes, not two different sets of modes as in Type II.
For Type I superstring theory, anomaly cancellation requires the gauge group to be SO(32).
[*] Type II does include open strings. However, they don't show up in the vacuum perturbation theory--they are only there in connection with non-perturbative effects (D-branes).