To start with, dispersion forces have many equivalent names. They are sometimes called induced-dipole induced-dipole forces, London Forces, London Dispersion forces, or van der Waals forces.
They are all names Chemists use to describe the same ubiquitous electrostatic attractive force. To have dispersion forces, a molecule must have electrons. Since all molecules have electrons, they all exhibit dispersion forces to some extent. Dispersion forces are induced-dipole induced-dipole forces that arise from fluctuation in the arrangement of the electrons around a molecule. Even non-polar molecules will not have a perfect distribution of charge for every instant of time. If there is a fluctuation that leads to the molecule having an instantaneous dipole, this dipole will induce a dipole in a neighboring molecule. This will in turn induce another in another neighbor. The induced dipole effect will propagate throughout the whole system. A measurement of this tendency is called polarizability and is given the symbol alpha, α.
As you look through the volume of an actual molecule on a point-by-point basis, dispersion forces will be quite small. However, since this force is everywhere throughout a molecule, the sum of all the interactions can be quite large. As a result, non-polar molecules with no dipole-dipole interactions can have much stronger IMF than polar molecules that have both dispersion and dipole-dipole attractions - so yes, size matters.
While not always "weak," dispersion forces are always very short-range. The energy for a dispersion interaction falls off as 1/r6. Thus the ability for molecules to pack tightly together has a huge effect on dispersion forces. Those that can get very close to one another have much stronger interaction than those that cannot. So once again, the three-dimensional geometry of the molecule has a great affect on its physical properties.
Finally, the polarizability has a larger effect on dispersion forces. The more electrons a molecule has and the farther they are from all the nuclei will affect how easily a dipole can be induced in the electron cloud (aka: polarized). This is why we perceive a molecular weight effect in boiling points. It is not that the mass of the molecules matters, it is simply that more mass implies more protons which implies more electrons. More electrons generally lead to more dispersion forces. Higher mass can also be farther down the periodic table. This will also be more polarizable. A great example of this is the diatomic halogens. F2 is a gas, Cl2 is a gas, but Br2 is a liquid, and I2 is a solid. All four have pure covalent bond and are completely non-polar molecules. But as we move down the periodic table the polarizability increases and thus the dispersion forces increase.