Frosts circle is a useful mnemonic for determining the molecular orbital diagram (relative energy of MO) for cyclic conjugated systems and for determining if a cyclic system is aromatic.
Lets see how we can construct the frost circle for benzene.
1) Inscribe the polygon (benzene is a hexagon) inside a circle such that every point on the polygon intersects the circle and one of the corners of the polygon is at the lowest point in the circle.
2) Add the molecular orbital energy levels at the intersections of the polygon and circle.
3) Erase the circle and the polygon then fill in your π electrons. Just like atomic orbitals you need to abide by Aufbau's principle (fill orbitals from bottom up) and Pauli exclusion principle (two electrons in same orbitals has opposite spin). Also just like atomic orbitals each molecular orbital can have a maximum of only 2 electrons.
These correspond to the 6 molecular orbitals that make up the π system of benzene. If each level is completely full and the structure is cyclic conjugated then it will be aromatic and exhibit 30kcal/mol delocalization energy and undergo substitution instead of addition. Take a look at the interactive Benzene MO's to get a feel for what the MO's look like.
1) Construct Frosts circle for the cyclopentadienyl anion, radical and cation, showing which are aromatic and which are not.
2) Construct Frosts circle for cyclooctatetraene. Is it aromatic?