The use of mantras isn't unique to Buddhism. Many traditions use some form of mantra practice. Practitioners of hatha yoga, for example, are well aware of the importance of the single syllable om in their practice.
In Buddhism, there are a plentiful number of mantras used, ranging from the simple above mentioned om, to the hundred-syllable mantra that some devotees recite hundreds of thousands of times as tool for purification. One of the most common Buddhist mantras is the six-syllable mantra om, mani, padme, hum. (Pronunciations are tricky with these mantras; this one usually is recited like this "Ome, mah-knee, pay-me, hung."
But a mantra that I have found very useful is a three syllable version: Om-ah-hum. In some practices, the recommendation is to utter (or silently recite) om in sync with your inhalation, hung on the exhalation, and, in the empty, quiet space between inhalation and exhalation, to observe the ah.
The simple mantra continues to reveal things to me the longer I use it, and over time I've found it to be a convenient grounding anchor whenever turbulence arises in my life. For me, the om and hum correlate to the yin and yang dualities, and can serve to represent any version of opposing energies that are present for you: pleasure-pain, good-evil, creation-decay. This revolving cycle of phenomenon is the way of the world, and the mantra is a way of reminding me of this basic truth of the rise and fall of all things.
Things get really interesting, though, in the "ah" between the "om" and the "hum." The middle syllable of the mantra, for me, represents a quiet openness, a non-judgmental spaciousness that serves as the ground, or matrix, in which all the om-hum phenomenon unfolds. It is, in essence, the syllable that represents pure awareness, which by its nature does not judge, but simply"knows" that phenomenon is being experienced.
Dwelling in the "ah" isn't a particularly familiar thing for us, because normally we are pretty much enslaved to the revolving cycle of our mental phenomenon, and really don't experience it with any kind of objectivity. We are either caught up in our feelings or emotions, or we're obsessed with our thoughts, and it never occurs to us that there's another quiet open state that is neither emotion nor thought. There is really no accurate metaphor for this spaciousness, although its sometimes said that glimpsing it is like suddenly recognizing the sky when you've spent all your time worried about clouds. When we do glimpse it, though, the feeling of relief is profound. Recognizing the "ah," and learning to relax and dwell there, is really what a spiritual path aims for.