I've been doing a bit of playing with various voip software lately.
Skype
Skype,
http://www.skype.com/, seems to be pretty nice. It is PC-to-PC, and will work best if you and your girlfriend have a broadband connection. Voice quality is pretty good. Its free (for now anyway), and seems to be built to overcome problems with firewalls, etc. There are a lot of TFP users who swear by it. Both PCs will need to have the Skype software installed.
MSN Messenger
I've had really good luck with MSN Messenger's built-in voice support for PC-to-PC calls. I prefer it to all of the others that I have used so far. Latency wasn't bad and both sides could talk simulatenously without cutouts. However, I've noticed that it tends to be finicky with some firewall/NAT setups. It has been my choice for PC-to-PC. I think its audio quality edges out Skype, although Skype is still very good. The built-in support for PC-to-PC calls in MSN Messenger is free.
Yahoo Messenger
Yahoo Messenger's voice support always seems to work, but I think the voice quality is crappy compared to everything else available right now. Sounds about as good as a CB radio to me. Y! Messenger voice is free as well.
Net2Phone
I've played around a little with Net2Phone. I haven't made any PC-to-PC calls with it, but the PC-to-Phone support isn't bad. I carried conversations out using it and I was fairly pleased. Your mileage may vary depending on the speed and latency of your internet connection. Also, don't waste your time signing up through MSN Messenger. My account was screwed up for a week because of problems between MSN and Net2Phone. I finally ended up speaking to someone in India who straightened the mess out. Just sign up directly at Net2Phone.com. I believe $10 is the minimum to open an account. Net2Phone bills by the minute for PC-to-Phone calls.
You download the software from their site and use a microphone and speakers (or headset) to call out. I highly recommend getting a decent headset. Using a microphone and speakers usually results in an annoying echo for the person on the other end.
I don't know that the savings per minute would be worth the hassle of using your PC instead of the phone, but I don't know how much a call to Argentina would be.