The Aufbau principle let's us build up an atoms electronic configuration by placing electrons into orbitals of every increasing energy. Hund's Rule tells us about how the electrons in an atom should be placed into degenerate orbitals. Technically Hund's Rules say that the ground state configuration should maximize the multiplicity of a configuration. What does that mean? The simple rule is that electrons should be placed into separate orbitals before going into the same orbital. This is because these best represent the configurations that are the lowest energy when we do the quantum mechanical calculation.
These pictures with up and down arrows for electron spin are a gross simplification of the realities of quantum mechanics. However, they do allow us to predict some measurable behavior for atoms such as the number of unpaired electrons. Thus Hund's rule has implications for predicting the properties of atoms as paired and unpaired electrons have distinct properties (in particular with interactions with magnetic fields).
Substances with unpaired electrons are called "paramagnetic" as they are attracted towards a magnetic field. Substances will all paired electrons are "diamagnetic" as they are repelled from a magnetic field.
Aufbau Principle and Hund's Rule