How do polarity and shape of molecules relate to the properties of a substance?

Polarity and shape affect the ability of molecules to aggregate and pack together.

Higher polarity, for example, may allow for hydrogen bonding to occur in certain instances which greatly increases a substance's melting and boiling points. If you compare CO2, which is non-polar and a gas at room temperature, to water (H2O) which is polar and is a liquid at room temperature, the different is due to polarity and hydrogen bonding.

A good example of the effect of molecular shape on physical properties of compound is a comparison between isobutane and n-butane. Isobutane has a boiling point of -11.7 degrees C whereas n-butane boils roughly around 0 degrees C. Why does n-butane have a higher boiling point? Because n-butane is a straight-chain hydrocarbon which can pack well with other n-butane molecules around it. This maximizes molecular interaction and the weak intermolecular forces (i.e. London/van der Walls) keeping the molecules together. Isobutane, meanwhile, has less contact with other isobutane molecules because of its 3D shape/geometry which is more rigid than that of n-butane.