Learn Russian: Можно присесть на этот стул?

The prefix при- here adds an "attenuative" (softening) meaning, expressing that the action is brief or small in scale. присесть means "to sit down for a moment" or "to perch". Using it instead of сесть makes your request sound more polite, casual, and less of an imposition on the other person.

Why do we use the Accusative case (на этот стул) instead of the Prepositional (на этом стуле)?

In Russian, the action of sitting down (присесть or сесть) is treated as motion towards a destination, so it requires the Accusative case to show direction (literally, "onto the chair"). If you were describing someone who is already stationary and sitting there, you would use the Prepositional case for location: Он сидит на этом стуле.

There is no word for "I" in this sentence. Can I add мне?

Yes, you can say Можно мне присесть?, using the Dative case for "to me". However, because можно means "is it possible" or "is it permitted", the pronoun is usually dropped in spoken Russian when it is completely obvious from the context who wants to sit down.

Could присесть mean something else depending on the context?

Yes! The prefix при- can also indicate an incomplete action. Because of this, присесть can mean "to squat" or "to bend your knees slightly". For example, in a fitness context, squats are called приседания. But when asking for permission near a chair, it always takes the attenuative "sit for a short time" meaning.