Molecular mechanics is completely parameterized, so as to reproduce the
stable conformations of common organic molecules and their relative energies.
Algebraic expressions (the
are used, which describe how
changes in bond length, bond angles, torsion angles, etc. affect the energy
of a particular structure. The parameters in these expressions are fitted
on experimental data (X-ray or NMR 3D structures, thermodynamics).
Therefore MM methods will yield very reliable results
for compounds closely related to the parameter set which was used for fitting.
Molecular mechanics is fast. It is applied in modelling programs, e.g. to clean up sketched structures.
Note that MM methods do not involve a description of the electrons and their behaviour. They are not meant to calculate properties that are determined by electronic effects.
The quantum mechanical methods attempt to calculate structures and energies,
at different levels of approximation of the Schrödinger equation.
Semi-empirical methods are partly parameterized. Not all electronic contributions are computed; some of them are replaced by parameters, chosen so as to reproduce preferred conformations, and to a lesser extent, the corresponding heat of formation. By far the most widely used semi-empirical 'hamiltonians' are AM1 (Austin Model 1, Dewar) and PM3 (Parameterization Method 3, Stewart), which differ in details. Both can be used in MOPAC.
Ab initio methods do not contain parameters, but can be used with increasing
accuracy of the description of all electronic contributions.
See for details the chapter on quantum chemistry in the computational chemistry tutorial.
The choice between any of the above methods is a matter of balance
between required accuracy and (computer)time.
For stable conformations of simple organic molecules, MM methods give good results. Quantum chemical methods are used for compounds where electronic effects play a role (e.g. conjugation), for structures outside the parameter set, and for non-stable configurations, like transition states.
If the semi-empirical methods yield unsatisfactory results (if possible there should be a check with experimental and/or literature data) one has to 'move on' to ab initio methods.