What defines a source of power, the current or the voltage?
Both walk together. Imagine voltage to be, as the name says, potential. In other words, is what your source can potentially causes in your circuit. It is its potential to generate a current. As the ohms law states (U=Ri), we can think of current being a consequence of a resistence connected to this source. This is almost a rule for almost every source of electricity in our days, but there are exceptions (below)! So, whenever you think of your wall outlet or a car battery or a cellphone battery, they have a voltage potential and the current will be calculated by which material or what are you connecting to the power source (by its resistance). Generally, the current specs tells us what is the MAXIMUM current that this power supply can handle. But it does not tell you that this source will be always +V volts and A amperes!

But note that this is the case for voltage sources. As the name says, it guarantees a constant voltage (ideally). So you calculate current because you know that the nominal voltage will remain the same.

So the excepetions will be current sources. Now everything is inverted! This kind of power supply guarantee's that the current will remain fixed. So you can calculate what voltage it is applying to your circuit so it can delivery that amount of current (also by ohms law) but generally we do not need that. Although a current source concept is very useful inside electronic devices, we are not used to see them in our days. But one good example is your telephone line comming from the wall. Those are current sources. Note that you cant damage the wires or the telephone company by making a short circuit to these wires. You can connect a multimeter and you will see that the current will be stable at some point (here in Brazil at 24mA). I can connect a 10Ohms resistance or a 300Ohms resistance and the current will be the same. Of course the voltage applied will be different, and that is how a current source works. 

So it all depends on what type of source you are dealing with. If it guarantees a current fixed, you can calculate the voltage difference between terminals. If it guarantees a voltage value, you can calculate its current depending on which load you put there. In most cases, those power supplies from computers, cellphones, etc are all voltage sources and its specifications guarantee a nominal voltage and a maximum current. But don't expect to have that current regardless of the connected load!