Does my computer have to be turned on and connected to the Internet?

Yes. As with any peer-to-peer (P2P) solution, there must be a network connection in order to operate. On the other hand, your computer does not need to be turned on for normal, landline phone calls.

General
  Introduction   Equipment
What is Skype?
Why use Internet Phone Wizard instead of a headset?
Why use Skype?
Does Internet Phone Wizard work with cordless phones as well as it does with corded phones?
Will making an Internet call have an effect on my computer's performance or ability to operate?
Will this product work on a Private Branch Exchange (PBX) phone?
Using Internet Phone Wizard
  Internet Calls Only   Internet & Normal Phone Calls
How do I setup Internet Phone Wizard for internet calls?
How is the sound quality?
How do I place a call?
Does my computer have to be turned on and connected to the Internet?
Can I unplug and replug Internet Phone Wizard into different USB ports on my computer?
What is the additional setup needed to make normal and Internet calls on my phone?
How do I switch between Internet and normal phone modes?
Can I make a normal phone call from a different phone (on the same phone line) when someone else is on an Internet call?