The only context in which I’ve heard of “conditional absolution” is if someone has incurred an undeclared automatic penalty reserved to the Roman Pontiff, and seeks pardon for it. That is, they’ve done something that automatically excommunicates or otherwise censures them, it was not a public act, and the Pope’s dispensation is required from the penalty. Let’s say someone desecrated the Eucharist. They confess it to me. I absolve them conditionally, since the penalty actually prevents them from receiving absolution absolutely (that was fun to say), and tell them to come back in two weeks. I write a letter to the Apostolic Penitentiary, the prelate who is empowered to lift censures in the name of the Pope, telling him that a penitent has confessed to that sin and I request the faculty to lift the censure due it. Two weeks later, I should have an envelope from Rome (they move very quickly), and inside it are two smaller envelopes, with a number printed on the outside. I open the letter in the presence of the penitent, while they open their copy. I read the letter, which decrees the lifting of the censure and imposes a penance. I take their copy and mine and burn them both, to protect the seal, and then I keep the envelope–that number is a reference number the Apostolic Penitentiary uses to follow up if need be.
