From Wikipedia article on Critique of Pure Reason:
1. Synthetic judgments add something to a concept, (e.g. an observation, like all objects are heavy) whereas analytic judgments only explain what is already contained in the concept (e.g. all objects are experienced by a subject.)
2. For Kant, mathematics is a priori synthetic judgment. For example, Kant uses the classical example of 7 + 5 = 12.
