Distinctions in Kant

A priori knowledge is knowledge that is not derived from experience but from concepts.  This knowledge does not depend on experience for its truth, and so experience can never disprove this kind of knowledge.  Common examples include mathematics and geometry.  1 + 1 = 2 will never be disproven by experience.  Just as mathematical propositions cannot be disproven by experience, neither can definitional propositions, such as <All bachelors are male>  (I use brackets here to indicate that I am talking about the content/sense of that proposition).

A posteriori knowledge is empirical knowledge gained through observation and experience.  This kind of knowledge cannot be gained simply by thinking about concepts; it requires interaction with the world and not just thinking about ideas.  Most fields of natural science will fall into this category, including zoology, ecology, cell biology, chemistry, pharmaceutical sciences, etc.  Many social sciences will also fall into this category, such as psychology, sociology and economics.  Certain axioms or principles used in these disciplines may be abstract concepts that qualify as a priori knowledge (if you believe in that kind of thing), but the methods of these sciences is to interact with the world and gain knowledge through observation and experimentation.

Analytic statements are statements where the predicate (verb, adjective, etc.) attached to a given subject (noun, name, etc.) is only presenting information about the subject that is already included in the definition for the concept of the subject.  For example, <All bachelors are male> is an analytic statement because part of the definition of bachelor is to be male.  <All lines consist of at least two points> is another example, because it is the definition of line that it consists of at least two points on a plane.  

Synthetic statements are statements where the predicate attached to a given subject is not already part of the definition of that subject.  In other words, synthetic statements amplify or add to our knowledge about the subject because they present new information that is not already included in the concept for the subject.  Kant thinks mathematical statements are of this kind.  He says that <7+5> does not already include <12> in the concept, so propositions such as <7 + 5 = 12> are synthetic.  Another example is <f=ma>.  The concept for <f> does not include <ma>.  The connection between the two concepts can be logically necessary, as it is the case for both of these propositions.  Yet the concepts are not one in the same, so the predicate is providing new information about the subject.