What does 'paid off' mean in the context of buying an iPhone?

"Paid Off" would normally mean that a user purchased the phone from a carrier like AT&T, Verizon, T-Mobile, etc or the phone was purchased directly from Apple and the user owns the phone free and clear.

The user either made monthly payments until the phone was paid off, or the user paid for the phone in full when it was purchased.

Of course, you really have no proof that the user really owns the phone unless they can provide written documentation.

Looking at the U.S. Apple site, I see various ways of paying for a new iPhone.

Buy

This means that you pay the full price of the iPhone up front, at the time of purchase.

Finance

This means that you pay for the iPhone in installments, over time. There are several types of financing:

• Apple Card Monthly Installments
• Apple iPhone Payments
• Carrier financing (AT&T installment plans, Boost Mobile Financing, T-Mobile Equipment Installment Plan, Verizon Device Payment Program)

All of these require you to have credit approval from whomever is doing the financing. E.g., Apple Card payments would only be an option if you had, or could get, an Apple Card credit card.

Note that if you choose any carrier financing option, the words "And your new iPhone will be unlocked" disappear from the order page. I think it is a sure bet that if you buy an iPhone with carrier financing, it will be locked to that carrier. You will be at the mercy of the carrier if you want to unlock the iPhone for use with other carriers. (They'll probably only do that once you complete the contract, if you are still their current customer.)

In skimming your post, I missed that you were probably referring to a used iPhone being sold by some private seller ("he type me paid off").

If you are thinking of buying a used iPhone, please read

The All Too Common SAD Reality of Buying a Used iPad/iPhone – Apple Community