I've come across this but yet to understand it. Can anyone explain in simple terms how a ii V I defines a key centre?
the V-I resolution is "satisfying" or easily processed by the human ear for reasons both according to the laws of physics (read up on the overtone series and the way we perceive sound) as well as our cultural lexicon (most every genre of western music uses it prominently, so if you have listened to western music all your life you have been 'trained' to expect and recognize it).
a very short explanation of he overtone series is this - any note you hear has a number of overtones, or basically shorter sound waves that get increasingly dissonant as they get smaller. the note at twice the frequency of the tonic is an octave - a very consonant tone. the third overtone (one third of the original wave) is the fifth - again, consonant. the overtones progress in this way - the next ones are the perfect fourth, major third, minor third and so on. if you think of this on a stringed instrument, it's very simple... fret the string in exactly the middle and you get an octave (12th fret on a bass). fret it a third of the way down and you get the 5th. a quarter of the way down gives you the perfect 4th. so the notes we perceive as the most consonant in music theory are closely connected with the actual length of the sound waves involved and the ratio of one to the other; the ear seems more able to make "sense" of certain simple ratios (1 to 1, 2 to 1, 3 to 1... as they get more complex the sound is more dissonant). in this sense all the notes of the major scale are present in the root note through its overtones, such that when we hear them played in other notes and chords, we have essentially already 'heard' them through the tonic and come to expect them. (this isn't a perfect way of phrasing it but you get the drift). this is why the circle of fifths works... in each case, the closest non-octave overtone is the fifth, so when you resolve from the fifth it sounds 'natural' or pleasing to the ear.
as someone else mentioned, the major second and major seventh of any major scale are both part of the V chord (say, the D and the B, respectively, in the key of C, are part of the G major chord). when a melody resolves stepwise to the root, either in ascending or descending fashion, either way the nearest note is part of the V chord, so it's very common to hear a melody resolve from, say, D to C over the chords of G and C respectively.
the ii chord, being the fifth of the fifth, resolves in a satisfactory manner into the V chord while also signaling to us that the V is not the tonal center, because if it was, the ii would be major. so it's a natural way to lead into the V without implying the V is the tonal center. (as an aside, very often you will see key changes preceded by a ii-V-I in the new key because it is a very palatable modulation between closely related keys; so for instance, if you were modulating from C major to G major, you might see the progression of A minor - D7 - G major. this provides a progression that is not too jarring to the ear - it starts on a chord that is common to the two keys and moves along the circle of fifths - while simultaneously giving a clear signal that the G is the new tonic.)