Was Borden able to do his transporting illusion because he had a "Prestige" of himself? I mean, he didn't have a twin brother. In the note he sent to Angier, it did read "Tesla". So was the Prestige twin the Pact he kept referring to? Or was it something else? But if he knew the Tesla secret, why was he so baffled by how Angier did his illusion? I mean, the apparatus was on stage, he should have plainly seen what Angier was doing. I'm confused.

Borden did not have a Prestige of himself, he did in fact have a twin brother that he concealed from the world. He even went so far as to falsify documents and photos to conceal the fact that his brother was his twin. Borden's real name was Frederick and his brother's real name was Albert. Put them together and you get Alfred, his given name throughout most of the book. So yes, his twin was the Pact that he kept referring to in his writings. Also, Borden did not know the secret of Angier's device built by Tesla. The device that Tesla built Borden was all for show - just lots of electricity to make his "illusion" look more dramatic. While Angier visited Tesla he remarked in his journals that he and Tesla were truly working on science whereas Borden was chiefly concerned with theatrics. That was why Borden was so confounded when he went to see Angier's act - he didn't even consider that it would be possible to transport yourself, only to make it look like you had been transported.

although PalmerJoss maybe you're right in terms of the book because in the movie they don't mention any Albert or Frederick as far as I can recall (I'll need to rewatch!)
My take is that Alfred Borden DID have a Tesla clone who he performed his trick with but the reason he couldn't figure out Angier's trick was because he couldn't fathom killing someone - even his own newly created clone - just for a magic trick.
I got that Borden figured out what Tesla's machine did when he was in Colorado and had his own double (the man who became Fallon) created..maybe by accident. Of course, he then had to falsify documents to give this clone an identity because Fallon was a clone and had been created out of thin air and therefore had no identity otherwise.
If they were twins they would essentially have to have been planning to do the trick their entire lives and both would have to be so totally dedicated to the idea of magic and this trick it's just a little too much. (I know a couple sets of identical twins and they're each very different people with different interests.) So Borden and Fallon were the same person cloned and from the moment they were cloned they each began to have different experiences leading them to grow into somewhat different people over the years and fall in love with different women...remember Fallon wouldn't have had that 'moment' with Sarah that caused Borden to fall in love with her. Also, it would just be too amazing a coincidence that he sends Angier to Tesla who just happens to have a machine that creates doubles handy.
My $0.02