Learn Polish: Oby skazany zrozumiał swój błąd w więzieniu. - May the convict understand his mistake in prison.

Skazany is a passive participle formed from the verb skazać (to sentence/convict). Literally, it means "the sentenced one." In this sentence, it functions entirely as a noun meaning "the convict." Even though it acts as a noun, remember that it still declines exactly like an adjective.

Why is zrozumiał in the past tense if the sentence is about a future wish?

The word oby (meaning "may..." or "I hope that...") contains the particle -by. Just like the word żeby, it requires the verb that follows to be in its past tense form (technically the l-participle). So, even though the wish is for the future, the grammar demands the past tense form zrozumiał instead of the future tense zrozumie.

Could we say jego błąd instead of swój błąd?

Not if you want to say he understands his own mistake. Because the convict (skazany) is the subject of the sentence, you must use the reflexive possessive pronoun swój. If you wrote jego błąd, a Polish speaker would assume he is understanding a mistake made by some other male.

How does oby differ from saying mam nadzieję, że (I hope that)?