What is the difference between ab initio calculations and molecular orbital theory?
Ab initio means calculations that are strictly from the principles of quantum mechanics. This contrasts with a non ab initio method which can involve parameters that do not arise directly from principles (i.e. results from experiment and/or 'better' calculations). One typical example of something that is not ab initio is a functional for Density Functional Theory. Because the exact functional is unknown, different functionals are generated using large datasets of high level calculations. These are often good at reproducing experimental data in many situations however you cannot derive them directly from the postulates quantum mechanics.
Molecular Orbital Theory (which can be implemented with the Hartree-Fock method) is an example of an ab initio method. One can (very roughly) come up with MO theory by approximating each electron as only feeling the effect of the average positions of the other electrons. So, if an orbital has an electron in it all the other electrons will feel a repulsion from the space occupied by that orbital. This results in a set of coupled one particle problems the solutions (eigenfunctions) which can be solved to find the Molecular Orbitals for the system.
More generally, ab initio calculations also refer to the more accurate post Hartree-Fock methods such as Configuration Interaction (CI), and Coupled Cluster (CC).