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What's the difference between a priori / a posteriori ?

For my philosophy final tomorrow, a lot of it is based off of Kant and Hume. I just need some clarification on the difference between analytic a priori, synthetic a priori, and synthetic a posteriori? I'm also curious as to why Hume and Kant both determined that analytic a posteriori is impossible. Thank you in advance!
gpt
A priori: knowable before experience (prior = before)

A posteriori: knowable after experience (posterior as in behind... which is sort of like after)

Analytic: found to be true by analysing the meaning of the concepts involved

Synthetic: found to be true by comparing the statement to the contents of the world
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Analytic and a priori is an easy match; "all bachelors are unmarried" is true by the meaning of "bachelor" without any need to go and examine any particular bachelors

Likewise, synthetic and a posteriori go well together; "that cat over there is black" is, if true, something you confirm by experiencing the cat to check that it is indeed black

Synthetic a priori is more difficult - it would need to be something that is true because of how it relates to the world, but doesn't require you to go and actually experience the world to check. Kant spent quite a while arguing that this was a thing that exists, and cited the example of certain mathematical truths, where the conclusion isn't evidently contained in the premises and tells us something useful about the world, but you can nonetheless prove it true without worldly experience.

Although, an expanded definition of analytic might throw the net around those propositions too, saying that if a mathematical proposition is provable from some set of axioms, then it is in fact contained within the definition of those axioms, just non-obviously.

Analytic a posteriori seems like a flat contradiction - if it's a posteriori then it requires experience of the world, but that directly implies that it's synthetic.