You can definitely receive international calls with the cell phones rgf mentioned. I am calling my boyfriend daily from the US on his cell phone which is exactly of such a type.
If I were you, I would buy in Spain one of the cheap cell phone packages- cell phone + prepaid calling card for 10.000 ptas in total (ca. 55 $). Even though you cannot use such a phone in the US afterwards, I still would guess that it is the cheapest solution. You can find offers where the phone comes with a prepaid calling card carrying exactly that value, and the idea of renting one can hardly be cheaper. I cannot guarantee that they have these offers right now since I am not in Spain at the moment, but I still saw them a couple of months ago.
They sell them in many places in the entire city, for example at the Internet Place at the Puerta del Sol (the Internet place is in the basement and on the street level they sell cell phones- you can easily find it walking around at Puerta del Sol. But as I said, there are plenty of other places, I am not aware of one at the airport, though. They do sell prepaid cards for cell phones at the airport, but I have not seen them selling phones proper.)
As long as you keep a positive balance on the card, the card remains valid for 9 months (these are the conditions of Movistar, I do not know about Airtel and Amena, but I think they are similar), and after that, they keep you the number for 3 months and you can still recharge it. This way, you can use the phone whenever you go to Spain. Movistar prepaid cards even have roaming in many other states, but of course, it gets expensive once you make or receive calls out of Spain.
The big advantage of cell phone contracts in Europe is that unlike US companies, they do not charge you as a cell phone user for incoming calls, the person who calls you pays the extra charges for air time (which makes calls TO cell phones rather expensive). As long as you remain in the country in which you have a contract, you just pay if you yourself call other people.
Therefore, if you basically use a cell phone to receive calls, these packages (cell phone + prepaid card) are a great bargain- for 10.000 ptas, you can receive an unlimited number of calls in Spain for a year, with the possibility to extend the validity of your card for another nine months whenever you recharge it.
And to make calls, you could mainly use public phones, as Nuria suggests, since international calls from your cell phone using prepaid cell phone cards are pretty expensive. Or you could go to a calling place where they offer slightly cheaper rates than from public phones (or at least that is what they advertise), for example the internet place at the Puerta del Sol I mentioned before.