The strength of a magnetic field is measured in units of Gauss (G), or alternatively, in Tesla (T). In the MKS (metric) system of units, 1 T = 1 kilogram*ampere/second^2 = 10^4 G.
For comparison, the magnetic field of the earth at the surface is on the order of 1 Gauss, where that of a Neodymium magnet is on the order of 10^4 Gauss. This means that Neodymium magnets produce magnetic fields tens of thousands of times stronger than those of the earth!
Technically, Gauss and Tesla are units of magnetic induction, also known as magnetic flux density. (This term is described in an earlier question.) Quantitatively, the force on a charged particle q moving with velocity v is given by the vector equation F = qv x B, where B is the magnetic induction.
Another common quantity of interest is the corecivity or corercive force of a magnet. Also measured in Gauss, the coercivity is the magnetic field required to demagnetize a material. For example, Neodymium magnets typically have a coercivity of about 12000 Gauss. Please note that the coercivity is the magnetic field required for de-magnetization. It is not actually a mesaure of the "strength" of the magnet, although the highly coercive magnets are usually quite strong.
The maximum energy product is used to determine the quality of magnetic materials. This is typically measured in Megagauss Oersted (MGOe - quite a mouthful!). The maximum energy product basically determines what materials make the best magnets.
Magnetic field strengths are measured with devices known as magnetometers, also called Gaussmeters.
Why are there so many odd units and terms used to describe magnetic fields? Keep in mind that magnetic fields appear any a large variety of contexts besides permanent magnets. Engineers and scientists regularly deal with magnetic fields in the study of electric circuits, motors, optics, and various other fields of technology. The power of the magnetic fields involved can vary a great deal, so many ways of measuring things have developed through the ages.