Learn Russian: Этот учащийся свободен от экзаменов.

It is a substantivized participle, meaning it is a verb form acting as a noun. It literally translates to "the studying one" (from the reflexive verb учиться, to study). Even though it functions as a noun meaning "learner" in the sentence, it declines exactly like an adjective, just with the reflexive particle -ся tacked onto the very end.

Since учащийся declines like an adjective, what would the feminine and plural versions be?

For a female learner, it becomes учащаяся. For plural learners, it is учащиеся. Notice a tricky detail here for B2 learners: the reflexive particle remains -ся even after a vowel (like the -я- in учащаяся). With normal verbs, you switch to -сь after a vowel, but participles always keep -ся.

How is учащийся different from студент or ученик?

Ученик usually refers to a primary or secondary school pupil, while студент specifically means a university student. Учащийся is a more formal, bureaucratic umbrella term that applies to anyone enrolled in an educational institution, which is why it fits perfectly in an official-sounding sentence about exam exemptions.

Why use the short-form adjective свободен instead of the long form свободный?