Richard: In folk music, Eve has pretty much hit it: the tonal center is the note around which the music makes the most sense. You consider all the cues--chordal and melodic progressions, melodic resting points, the frequency with which particular notes are used or stressed, etc.--and from that intuitively deduce the tonal center. It's a matter of pattern detection over a whole phrase or tune rather than hard rules. Fortunately, our intuitive musical sense is far more acute than our theoretical understanding, so usually, the tonal center becomes apparent rather quickly--you just "hear" it.

The intervalic relationships from the tonal center to the other notes determine the mode. Typically, you'll detect the tonal center and the mode at the same time, though your initial guess of mode may be mistaken until more of the tune unfolds: aha, that seventh, flatted, means the mode is Mixolydian instead of major/Ionian, as I first assumed! Tunes can also shift modes and/or tonal centers (such as a vascillating seventh, a switch from major to minor or a temporary modulation up a fourth). Gapped scales can leave mode indeterminate, although accompaniment may favor a particular modal interpretation. And I'm still talking about traditional folk music, not modern extensions into jazz chords, frequent modulation, chromaticism, polytonality and the like.

And now (we hope), back to the hunt for tunes in Lydian or Phygian...