The simplest molecule to understand is molecular hydrogen, H2, because it is a combination of the simplest atom.
What do we get from the quantum mechanical solution to the hydrogen molecule using MO theory? Just as with quantum mechanical calculations for atomic system, the MO calculation yields orbitals (wavefunctions) and energies. The electrons can then be placed into those orbitals starting at the lowest energy orbitals with two electrons going into in each orbital. A key difference is that now we must choose a geometry at which to perform the MO calculation. For H2 the geometry is defined by the separation between the two hydrogen atoms.
To examine the "bonding" we can compare the molecular orbitals (MO) to the atomic orbitals (AO). We need to make this comparison at a particular distance, since the MOs depend on the atomic separation. Generally we will look at the MOs at a separation that minimizes their energy (the bond length). We can represent this comparison by the following diagram.
Here we show the MOs in the middle and the AOs on each side of the diagram. Energy is lowest at the bottom of the diagram and increases moving up. This particular diagram shows the orbitals for both the hydrogen atom and the hydrogen molecule. The AOs are the 1s orbitals from the hydrogen atom. The MOs, in the middle of the diagram, shows the two lowest energy MOs. Each hydrogen atom has one electron. For the atomic system we put one electron into each hydrogen (one atom is depicted on the left side and the other on the right side). This can be compared to the molecular system by placing these same electrons into the MOs. There are two total electrons, so we put these same two electrons into the MOs. Now both electrons can go into the lowest energy MO (the one label \(\sigma\)). This orbital is lower in energy than the AO orbitals. This means the energy of the molecule (the two electrons in the sigma orbital) is lower in energy than the separated atoms (one electrons in each 1s AO). The molecule is more stable than the atoms. Thus we have a chemical bond between these two atoms (they are lower in energy together than apart).
Moreover, the MO calculation is quantitative, and we know not only that it is lower, but how much lower and at precisely what distance! Not all the MOs are lower in energy than the AOs. You can see the one labeled \(\sigma\) is lower while the one labeled \(\sigma *\) is higher in energy. MOs that are lower in energy than their corresponding AOs we call "bonding". MOs that are higher in energy than their corresponding AO's we call "anti-bonding". If the MOs happen to have an identical energy (or very similar energy) to the AOs we label them as "non-bonding".