I will explain its role in solving Diophantine equations defining curves. Then, the height of a point is like the degree of some curve in a polarized variety. There are many ways to add another dimension to the projective space.
Different ways give different heights.
For the lecture notes as a .ps or .pdf file click here. This includes, for example, an abelian variety. Finally, I will explain consequences of using successive minima relations. This includes a Manin-Mumford type conjecture and an equidistribution theorem on Galois orbits of preperiodic points with respect to the invariant measures on Julia sets. For the lecture notes as a .ps or .pdf file click here.