Learn Finnish: Kissa istuu pöydän reunalla. - The cat is sitting on the edge of the table.

Technically, reuna is a regular noun meaning "edge", but in this sentence it acts exactly like a postposition. Just like true postpositions (such as päällä or alla), it requires the word it modifies to be in the genitive case, which is why pöytä becomes pöydän.

How would this sentence change if we used the lesson words asti or saakka?

You would use asti or saakka to show movement reaching a limit. Because istuu is a static verb (sitting in one place), we do not use them here. However, if the cat was walking all the way to the edge, you would use a motion verb and the allative case for movement onto a surface: Kissa kävelee pöydän reunalle asti.

Why is the adessive case -lla used on reuna?

The adessive case (-lla or -llä) expresses being on an outer surface. Since the cat is sitting on top of the edge rather than enclosed inside of it, we use reunalla. If a nail was driven into the edge itself, you would use the inessive case instead: reunassa.

Why does pöytä change to pöydän?

This is a reminder of standard consonant gradation. Adding the genitive -n ending closes the final syllable of pöytä. This triggers the weak grade, softening the strong t into a weak d.