A synthetic proposition is a proposition that is capable of being true or untrue based on facts about the world - in contrast to an analytic proposition which is true by definition.
For example, "Mary had a little lamb" is a synthetic proposition - since its truth depends on whether or not she in fact had a little lamb.
The truth or falsity of synthetic propositions is contingent - their truth depends on what the universe is like.
Are there synthetic propositions known a priori?
Whether or not it is possible for a synthetic proposition to be a priori is a matter of considerable controversy in philosophy.
The empiricists hold that there are no synthetic a priori truths.
The rationalists hold a contrary view - that there are synthetic truths that are also a priori.
In rationalist view, certain concepts are derived from experience - but once these concepts have been derived, they are seen as necessarily true.
For example, in this view, we derive the notion that 2 + 2 = 4 from experience - it would not have occurred to us had we not experienced two things, and another two things, becoming four things - but once we do experience it, we recognize it as a necessary truth.
The Kantian view
Immanuel Kant's view was more involved.
In his view, our knowledge of the world is limited to the phenomenal world - the world as it is known by the mind. The noumenal world - the world as it really is - is not knowable. The nature of the phenomenal world is constrained by the nature of the human mind.
So, certain fundamental truths about the world - for example, that if A precedes B and B precedes C, A must precede C - are truths about the phenomenal world only. We can't prove "A must precede C" in the noumenal world. So such fundamental truths are not universal truths - and thus not analytic. However, they are a priori - in the sense that our minds could not possibly think otherwise.