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You will probably not find a better explanation than that of the Dalai Lama.
It is very good to recite the mantra OM MANI PADME HUM, but while you are doing it, you should be thinking on its meaning, for the meaning of the six syllables is great and vast. The first, OM, is composed of three pure letters, A, U, and M. These symbolize the practitioner's impure body, speech, and mind; they also symbolize the pure exalted body, speech and mind of a Buddha.
The entire talk can be found here: http://www.sacred-texts.com/bud/tib/omph.htm
Thus the six syllables, OM MANI PADME HUM, mean that in dependence on the practice which is in indivisible union of method and wisdom, you can transform your impure body, speech and mind into the pure body, speech, and mind of a Buddha. It is said that you should not seek for Buddhahood outside of yourself; the substances for the achievement of Buddhahood are within. As Maitreya says in his SUBLIME CONTINUUM OF GREAT VEHICLE (UTTARA TANTRA) all beings naturally have the Buddha nature in their own continuum. We have within us the seed of purity, the essence of a One Gone Thus (TATHAGATAGARBHA), that is to be transformed and full developed into Buddhahood. (From a lecture given by His Holiness The Dalai Lama of Tibet at the Kalmuck Mongolian Buddhist Center, New Jersey.)
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Mani literally means 'jewel' in Sanskrit, and it is used to mean Buddha nature, for instance in the metaphor Indra's Net. It is described as a possession or attribute or treasure of awakened beings.
Padme is Sanskrit for 'lotus flower'. It is a powerful symbol in Buddhism, because the flower rises from the lake mud and is held above the lake, even while it is non-dual, not separated from the mud. Even in this mundane life, we can rise above, and attain purity.
Om & hum have been called 'interjections without linguistic meaning, but widely known as divine sounds'.
So I would suggest the mantra as a whole is directed at remembrance, invocation, of the powerful Buddhist symbols of jewel and lotus, of intrinsic power of minds to awaken symbolised as faceted jewel, and dynamic power to awaken symbolised as a lotus flower that rises above what is impure.
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Complete reference to this can be found in Wikipedia: