iPhone headsets have both microphone and headphone functionality using 1 jack. However, my computer has two jacks for two different things: Microphone and headphones. How can I use it for both on my computer?
You should be able to find an adapter that has two plugs for your computer.
Here's one I found by googling, but I can't vouch for the site myself: http://www.showmecables.com/viewItem.asp?idProduct=8183#
The iPhone headset jack is TRRS, so it's a total of 3 channels (1 in, 2 out) with a common ground. This web page has the pinout and some other information: http://geekspeak.org/blog/2008/08/12/iphone-headset-plug
(TRRS = LeftRightGroundMic)
Someone may know of a breakout box or cable, but you're probably going to need to know some electrical engineering to solder a solution for yourself.
It would be much less hassle for you to buy another headset for your computer. Analogue headsets are extremely cheap on eBay, although I would recommend that you spent a bit more on a USB headset (maybe Plantronics or Logitech) if you are going to use it a lot.