Can someone please answer this question for me. Is the key of A on the fifth fret, with standard tuning and no capo? The reason I ask, is that someone told me it was in a different place, which I can only imagine being the case if a capo was used.
The 'key of A' is all over the guitar fingerboard! The key of A comprises seven specific notes, all of which 'fit' into the scale of A major.
I think what you're asking is where is the note A. Again, there are several places on the fretboard where an A lives.
The lowest one is on the bottom (fattest) string, 5th fret. The same note is also played on the 5th string open. The octave of that, also A, is found on the 4th string, 7th fret, and 3rd string 2nd fret. The only other A is on the 5th fret top (thinnest) string, an octave higher again. That may be the one referred to. There are several other frets/strings where you can find an A note as well.
When a capo is used, it works like your fingers, on the fret it's placed on. All the other, higher notes will not be affected, and will still have the same names.
As a complement to Tim's answer, here's a possible interpretation of the question. Suppose that you have only learned a few chords, maybe E, A, and B7. You could then play quite a few simple songs but only in the key of E. Now put a capo on the first fret and play the same fingerings (up one fret) and you will get F, Bb, and C7 suitable for the key of F. Put the capo on the fifth fret and you will get A, D, and E7 suitable for the key of A.
However, this is a special case, if you know, or can figure out, the fingerings then you could play in any key with the capo in any position.
Edit: up / down flipped based on comments.
As @Tim says, the 'key' of A is all over the fretboard as it incorporates several notes. The locations of the note of A however is shown here:
It's generally a good idea to memorize the notes on the top 2 thickest strings (6th string and 5th string - the last two rows in the table) when beginning as it helps you identify what you are playing as you improve.