As you seem to already be clear on, one cannot tell what key a piece is being played in by looking at the key signature, because every key signature is used by two keys, one major key and one minor key, which are said to be relatives. As this signature has no sharps or flats it is either CM or Am. Because both keys use the same signature one must then look at the music itself. Often a piece will start and end with a triad chord, such as CM (CEG) or Am (ACE) which will nail the key. This piece opens with two A's (an octave apart), but as John points out, continues with melody lines as opposed to a chord. Still, starting with two A's strongly suggests that the piece is written in Am as opposed to CM.
To further complicate matters, there are three minor keys. Am natural has no sharps or flats. Am melodic sharpens the 6th and 7th notes (F & G) when ascending but reverts to the key signature when descending. Finally, Am harmonic sharpens the 7th (G) throughout while leaving the 6th untouched. You will notice in this piece that G is never played natural but rather is always sharpened whether assenting or descending while F is always left natural. Hence, again as John notes, the key is clearly Am harmonic.
Alas, I am digging into a shallow knowledge pit here having never studied music theory. I do know for example that GM is related to CM and as GM sharpens the F you might find a piece where the key signature is that of CM (and Am) and the key is CM but it wanders off into GM for a spell, and indeed Delcamp give us an example in D02, lesson 5, http://www.classicalguitardelcamp.com/v ... f=41&t=670 so without knowing what is happening one might be initially puzzled by the presence of F#, but I digress. Hopefully my explanation is basically sound and/or someone will correct me.