Is it possible to incorporate Bitcoin into a app my company is developing in order to process person to person payments?
You can build your own inner ecosystem based on bitcoin's block chain where every transacion is in bitcoins. Then you can make a gateway for outer payments using one of the existing commercial payment processors (InPay, BitPay, BitInstant etc).
Moreover you can create your own cryptocurrency using bitcoin's open-source code. But then you have to take care of minig process by yourself and create many additional tools (wallet, minig software, blockchain API). You can also have problems to exchange your own cryptocurrency with bitcoins or dollars. So if you need connection with the outer ecosystem it will be better to use bitcoins then create your own software.