A proposition is analytic if true or false in virtue of its meaning only. The contradiction of an analytic truth is nonsense. Example: red is a colour. Bachelors are unmarried.
It is synthetic if true or false in virtue of the world. The contradiction of a synthetic truth is meaningful (albeit false). Example: human blood is red. John is a bachelor.
It is known a priori if you don't need experience to know its truth value (example: math and conceptual analysis), a posteriori otherwise (scientific truth, facts).
Intuitively, analytic and a priori seem to go together, and synthetic and a posteriori as well. You don't need experience if the meaning only is at stake, otherwise you do need input from the world. Kant however assumed that some mathematical and metaphysical statements are synthetic a priori, a priori because they are known by intuition only, yet synthetic because their contradiction is not absurd. Example: the axioms of euclidean geometry. One can formulate consistent non-euclidean geometries, but Euclid's axioms are true in virtue of physical space and known a-priori (because according to Kant, space is a condition of experience).
This assumption was challenged afterwards (notably, euclidean geometry is not the geometry of physical space, so math axioms might be pure linguistic conventions).
Finally Quine challenged the analytic synthetic distinction on the ground that one cannot distinguish clearly the linguistic and factual components of a sentence. Quine believed there is no such thing as a-priori meaning.
A third important, related dichotomy is necessity / contingency. Traditionnaly, empiricists conflate analycity and necessity but Kripke challenged this (he assumes some metaphysical necessities are synthetic, such as gold's atomic number).
I've never heard of analytic a posteriori, although Kripke gave examples of analytic contingency, such as the choice of a conventional measurement unit.
