Everything has a vibration. Every object, every living thing, is constantly vibrating. If you could record the sound of every gross and subtle object that had ever been, vibrating at all of their frequencies, and then played all of those recordings at the same time, the resulting sound would be Om. So when we chant Om, we are joining with the vibration of the universe, the most sacred sound.
Om is the sound of the interconnectedness between all of nature, all of humanity. It is the bija-mantra (seed mantra) for the crown chakra, connecting us to the source of Creation.
To chant Om correctly, we actually chant 3 sounds: A (ah), U (oo), M. You may see Om written as AUM for this reason. Each of these sounds has a meaning. To the Hindus, this is the name of God, each letter representing the divine energy of Shakti: Brahma Shakti (creation), Vishnu Shakti (preservation), and Shiva Shakti (liberation/destruction).
The symbol for Om (as depicted above) is not a phonetic representation of the sound, but rather a picture of the four states of consciousness. The bottom loop on the left represents a waking and thinking sate (jagrat). This is the A of AUM, beginning, birth, represented by the body. When we are in this state beta brain waves are present. The loop on the right represents dreaming (swapna), which is likened to being asleep and thinking. This is the U of AUM, preservation, life, the middle, represented by the soul. When we are dreaming the brain is in theta waves. The top loop on the left represents dreamless sleep (sushupti), or sleeping without thinking, the M of AUM. This is death, destruction, winter, represented by the spirit. Here our brain moves into delta waves. There is a swoosh that is the veil of illusion (maya), separating these three states from the fourth state, which is represented by the dot at the top. This state is pure consciousness, or being awake without thinking (turiya). This is unchanging, bliss, stillness, unity, and is found in the silence after the AUM. When we are present in this state our brain moves into alpha waves.
Swami Sivananda said, "Turiya, or the fourth state, is that in which the individual soul rests in his own Sat-Chit-Ananda," or the bliss that is found in the wisdom of being present. How do we find this place? How do we come to a state of being without thinking? One of the ways is through meditation. There are many ways to meditate, but one of my favourite is called Yoga NIdra. Yoga Nidra is a guided meditation that begins by connecting you to your body, and then to your breath, and then to your consciousness. If you are interested in experiencing Yoga Nidra, I have uploaded a 20 minute audio track that you can download and listen to. It's just me talking in my room, nothing fancy, but it is here for your personal use.
Om is the bringing together of all of the four states of consciousness in one symbol, in one sound. It is the union that is yoga. And as Richard Miller said, "Yoga is the bringing together of that which was never separate."