These three terms refer exclusively to the time of payment, not to the form of payment. They mean, respectively, at the time of executing the sale contract, subsequently or beforehand. There are many possible ways of expressing them, but I don't know of an exactly equivalent legal formula in English.

The original of the source Justin found in wordreference is this:
" El Derecho Civil distingue tres formas de pago del precio por razón del tiempo en que se haga:
1) Confesado, si se ha satisfecho antes de la formalización del documento de que se trate.
2) Al contado, si se hace en el mismo momento de la firma del contrato.
3) Aplazado, si se conviene para ser pagado en un momento posterior y, por tanto, nace la obligación del adquirente de satisfacerlo."
http://www.notariosyregistradores.com/doctrina/ARTICULOS/200...

Let's take them in the order they appear in your document:
Al contado can mean in cash, but it doesn't here. It has two meanings:
"1. loc. adv. Con dinero contante.
2. loc. adv. Con pago inmediato en moneda efectiva o su equivalente. U. t. c. loc. adj. Pago al contado."

Meaning (1) is "in cash", but meaning (2) is "on the spot". The relevant word is "inmediato". It says "en moneda efectiva o su equivalente. Cash or cash equivalent. Not an IOU, but a banker's cheque or any other negotiable instrument would do fine, for example. As I say, it refers to time, not form, of payment. "In cash" is wrong here, because for example a payment in cash two weeks later is not "al contado" for these purposes.

And our source says it means "en el mismo momento de la firma del contrato", so there you are. You could translate it "at the time of executing the contract" if you like.

"Aplazado" doesn't mean "in instalments". Instalments means that payment is made in stages, not all at once. But "aplazado" simply means later, some time after the contract is signed. It could perfectly well be that the whole price is paid in one go, not in instalments. As our source says, simply "en un momento posterior".

And yes, "confesado" has nothing to do with stated or declared or acknowledged. It simply means that payment has already been made and received before the contract is executed: "se ha satisfecho antes de la formalización del documento". It doesn't mean a promise to pay, it means actual payment. It's correctly explained here:
http://forum.wordreference.com/threads/precio-confesado.1760...

Alejandro's answer here is correct; all the other answers, alas, including the chosen one, are wrong:
http://www.proz.com/kudoz/spanish_to_english/law_contracts/3...

Confesado is correctly handled here:
http://www.proz.com/kudoz/spanish_to_french/law_patents/6222...

And by Andy here, though his translation doesn't fit in this context:
http://www.proz.com/kudoz/spanish_to_english/law_patents/456...

And by the way, Kirsten is quite right about "confesado" in the TC question. It does not mean "acknowledged". More guesswork based on dictionary definitions, I'm afraid. Most of the answers for "confesado" in the KudoZ glossary are also wrong guesses and quite worthless.

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Note added at 2 hrs (2015-12-29 16:11:40 GMT)
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I think the origin of "confesado" is probably the idea of payment having been received and receipt acknowledged. In the old days Spanish receipts used to say "Confieso que he recibido...".