Updated: Apr 6, 2021
What is a mantra? The New Oxford Dictionary defines mantra as a "noun denoting a word or a sound repeated to aid concentration in meditation." The definition adds that a mantra is given to a trainee meditator by their spiritual teacher.
That little addendum is both correct and somewhat incorrect. Indeed, spiritual teachers sometimes prescribe a mantric practice to some meditators in their charge. Still, a teacher's specific mantric prescription and/or imprimatur is not required to pursue a mantric approach to meditation.
While the mantra derives from very early South Asian spirituality, the word has become ubiquitous outside its spiritual connotation. The New Oxford Dictionary offers an example: "[T]he environmental mantra that energy has for too long been too cheap." A comedian, whose name I've forgotten, equated mantra with yadda-yadda in some not-so-funny bit.
Finally, the Sanskrit definition: "[L]iterally 'a thought, thought behind speech or action', from man-'think,' related to mind." Hmm, I'm not sure how much that helps, but we strive to be thorough.
Now and again, I survey websites and apps dealing with meditation as well as what is commonly these days labeled "mindfulness practice." While not a perfect separation, I've noticed that much of the former derive from spiritual organizations or is presented with a religious foundation. The latter--this is not a criticism, merely observation--tends to instruction and practice void of reference to an underlying tradition or history. I see the latter as a "to-go" or "just do it" methodology. Exactly how I began my practice many years ago.
To finish my judgment and move on, we at CloudMeditation.com believe firmly in "just doing it." We stress that the practitioner is ultimately his/her own spiritual teacher and will eventually pursue the origins and traditions of their method to the depth of their interest.
Now, to the point of this article: Often in both types of sites or apps, one finds the superstar of mantras--OM (also traditionally spelled AUM, or A-U-M.).
So what? In the four to five thousand years of Indian spiritual development, hundreds of mantras have likely been devised. Consider India's spiritual influence on later traditions, one may add to the number. Buddhism and some have asserted ancient Christianity incorporated intoning sacred words or phrases into their practices.
I worked in a spiritual group with a Buddhist priest from a different tradition than mine. Before our virus troubles, I frequently attended meetings in a temple of his sect just a mile from my home. Namu-amida-butsu is their formal mantra. Nam-an-da-bu is an abbreviated version. Meaning: "I take refuge in Amida Buddha." Amida is one of the many symbolic incarnations of the Cosmic Buddha. Japanese Buddhist Master Shinran, the Shin sect founder in the 1200s, said, "Just say the Nembutsu and be saved by Amida. Nothing else is involved." 
Over the years, I've run across the assertion that the Christian "Amen" may derive from mantric tradition. And yes, I've heard it intoned as one. Try it: Ahhhh . . . Mennnn. . . Stretch both syllables out and into silence for as long as you are comfortable.
Out of myriad mantras that most people do not know like Nam-Myōhō-Renge Kyō, or the somewhat better known Om mani padme hum, the simplest of all is known by most everyone--OMmmmmmmmm. . . 
When I teach this method, inevitably, someone will assert the simplicity of the technique. Heck, anyone can grasp the practice in an instant. Implicit in the assertion is a sharp criticism of my value as an instructor, likewise the mantra's worth. Something so simple simply cannot be of much value. Countering these two objections is precisely the purpose of this article.
OM is simple but hardly trivial. As I've said, the practice has been instructed and discussed in written form for several thousand years and likely existed as oral teaching far longer. For our purposes, I'll acquaint you with two sources, one you know if you visit CloudMeditation.com and one that will be new to you. Both are two of the briefest and most accessible scriptures in the Vedic tradition. Both are "how-to" manuals, simple to grasp, easy to put into practice.
The first is our old friend, the Vigyan Bhairava Tantra, methodologies to experience the Divine. Within its 112 simply instructed contemplative methods, several relate to sound, listening, or intoning. OM is there, verse 42. Here is one translation:
"O Goddess, chant AUM, etc., slowly. Concentrate on the void at the end of the protracted sound. Then with the supreme energy of the void, one goes to the void." 
Sanskrit is a spiritual language. While I'm not a Sanskrit adept, I've known and communicated with some who are. Sanskrit is very often intoned rather than simply spoken. Each letter and its sound is accorded spiritual significance. Inherent in its use is the equal importance of sound and silence.
Traditionally, OM is written A-U-M, three letters. While we don't intone ah-oo-om, the three syllables are indeed there. Hearing, using our body to feel the vibrations and sense the subtle movements of it as one progresses through the full intonation, is part of the practice's minute observations.
A fourth letter is implied in addition to A-U-M, the absence of sound at the end of intoning. Sinking into and resting gently in the silence before a natural inhalation is a crucial part of the technique. As one settles into the practice, intonation and the companion silence lengthen and deepen. What sounds like one sound, one silence, or three subtle sounds, and one silence are genuinely four. And that takes us to our second ancient, profound, yet brief and utterly accessible text on OMmmmmmmmm. . .
Vedas are the oldest Sanskrit spiritual texts in Indian tradition, the earliest thought to have been written down 4-5,000 years ago from earlier oral sources. Upanishads are later writings of the tradition generally thought to have been written down somewhere around 200-400 BC.
While AUM is mentioned in several of them, the shortest one, the Mandukya Upanishad, is all about its meaning and significance. The text begins: "All is AUM...The whole universe is the syllable AUM. Everything that was is, or will be is, in truth, AUM. All else which transcends time, space, and causation is also AUM."
The Mandukya Upanishad proceeds to describe a four-fold structure of A + U + M + "silence" from several perspectives. Here are some brief samples:
Time: AUM+silence is the past, present, and future, as well as that which transcends time--the Divine, God, or Absolute Reality, you fill in your own word.
Self: AUM+silence enumerates four states of consciousness: waking, dreaming, deep or dreamless sleeping, and the consciousness of oneness with the Divine [ekatma]. The lesson of the last is, THAT thou art! God and you are One.
Body: The waking consciousness is the gross body, outward-looking and knowing. The dreaming consciousness is the subtle body, inward-looking and knowing. Think of the whole universes of people, places, dramas, and what else we create from inside ourselves when we dream. Deep sleep (dreamless) consciousness is the causal body. Our consciousness is undistracted. Finally, the fourth consciousness is body-less. It transcends the three common consciousnesses. One may melt into or meld with the Divine free of absolute or relative conceptualizing because, in Ultimate Reality, one was never separate from it. That perceived separation was one's own deception of oneself. Buddhists call this body-less experience the experience of Emptiness or Nothingness. But, absolutely EVERYTHING is included in this NOTHINGNESS. Ultimately, EMPTINESS is utterly full of EVERYTHING.
In simple summary, the four consciousnesses are:
Relative outward consciousness.
Relative inner consciousness.
Non-relative spiritual consciousness.
Unity with the Eternal. 
All the egghead stuff aside, if you haven't, give OM a try. If you are reticent to intone vocally, or someplace where it might be inappropriate, intone silently, internally. It's just as effective. Personally, I intone audibly and inaudibly another mantra that's commonly written Ham-Sah. The first syllable is just a rasp on the in-breath. The second is sounded, or not, on the out-breath. As with OM, the exact instructions and principles of the mantric practice apply.
I owe a great debt to Swami Sarvapriyananda, a sanskrit and Vedic scholar, yet a tremendously accessible and charming teacher. I've put a couple of YouTube links below. 
OMmmmmmmmm. . .
 English: "Devotion to the Mystic Law of the Lotus Sutra [Scripture]/ Glory to the Dharma of the Lotus Sutra." This mantra is chanted within all forms of Japanese Nichiren Buddhism based on the teachings of the 13th-century Japanese Buddhist priest Nichiren (1222–1282)
Also written as Auṃ maṇi padme hūṃ.
 Chaudhri, Ranjit. Vigyan Bhairava Tantra: 112 Meditations for Self Realization. Prakash Books India Pvt. Ltd, 2008
Please note this is just one translation of verse 42. Other translations differ widely, but all instruct intoning sacred sounds.
 Charles Johnston, The Measures of the Eternal - Mandukya Upanishad Theosophical Quarterly, October 1923, pages 158-162
 Secret of OM: Swami Sarvapriyananda.
The Ultimate Secret of OM: Swami Sarvapriyanada.