Synthetic judgements are in a way the opposite of analytic judgements. Analytic judgements are judgements which are made within the limits of what is known. If, for instance, someone had determined that for plant p to grow three things are necessary: dirt (d), water (w) and sunlight (l), I would be able to analyticly conclude that if p would not grow, but wither, and d and l are in abundant supply w is absent. Synthetic is the opposite of that in the manner that we look at the given facts, but do not know the answer yet and so cannot pick it out of an available set of answers. If we would know what 7, 5 and + means we could conclude synthetically that 5+7=12. This without knowing 12 on forehand.

A priori is perhaps best understood when contrasted with a posteriori. A posteriori means after perception and a priori means before perception. Kant speaks of this in the context of brain functions. Kant is what people would call a rationalist, which means that he believes that to understand our perceptions we need to know how to deal with them a priori. Empiricists believe that we percieve and on the bases of our perceptions we learn to think. Rationalists give empiricism a place a posteriori. In a rationalist view one knows what one percieves (in a basal way) a priori, and works out details (reasons) a posteriori.
A priori are what Kant calls "intuitions"; which he has divided into categoria. All categoria are based on a basal understanding of space and time

Here is great straight forward resource:

Kant: Judgments