The only way I know how to send Bitcoin is through an exchange. How would I do this from one computer to another without using a third party. Is this even possible?
Just simply connect your PC on the internet and make a web wallet, and then send your bitcoin, simple as that. Sending bitcoin to an exchange is only when you are going to use your bitcoin on Trading, to buy some Altcoins that you could Trade in the long run or Altcoins that is promising that could assure you to make some profits in the long run.
E D I T : I'm still confused on what you mean, so please correct me if I got it wrong.
I think thats all you explain is that his mean.
However, dont ever use exchanger to store your bitcoin. Use exchanger only for trading.
The remain coin store in own private key wallet if you dont use it for trading