Mantra #1 ~ "OM"

"Mantra" (Sanskrit मंत्र - «m.; also n. mantram» consists of the root man- "to think" and the suffix -tra, designating tools or instruments, hence a literal translation would be "instrument of thought") means a sacred utterance, numinous sound, or a syllable, word, phonemes, or group of words believed by some to have psychological and spiritual power. In other words "Mantra"  has two parts: man, which is the root of the Sanskrit word for mind; and tra, which is the root of the word instrument.
Jan Gonda, a widely cited scholar on Indian mantras, defines mantra as general name for the verses, formulas or sequence of words in prose which contain praise, are believed to have religious, magical or spiritual efficiency, which are meditated upon, recited, muttered or sung in a ritual, and which are collected in the methodically arranged ancient texts of Hinduism. 

There is no universally applicable uniform definition of mantra because mantras are used in different religions, and within each religion in different schools of philosophy.

Mantra may or may not be syntactic or have literal meaning; the spiritual value of mantra comes when it is audible, visible, or present in thought.

They are typically melodic, mathematically structured meters, resonant with numinous qualities. In more sophisticated forms, they are melodic phrases with spiritual interpretations such as human longing for truth, reality, light, immortality, peace, love, knowledge and action. In other forms, they are literally meaningless, yet musically uplifting and spiritually meaningful.

Mantras were first written about in the ancient Vedic scriptures thousands of years ago - going back as far as 1000 B.C.. At that time, mantras were simply either a word or a collection of words or sounds aimed to be repeated to attain positive changes in life.

The most basic mantra is "Om", which in Hinduism is known as the "pranava mantra," the source of all mantras.

"Om" is a Sanskrit word that is meant to convey the “original vibration of manifestation” – in other words, when intention manifests in the physical realm.

 According to the Vedic tradition, the ancient sages were able to hear the subtle vibrations produced by everything in nature―the sounds of the wind, thunder, butterflies, rushing rivers and all other creations. They recognized that these sounds are the manifestation of spirit into matter. They identified “Om” (or aum) as the most elemental sound, representing the infinite universal consciousness.

The Hindu philosophy behind this is the premise that before existence and beyond existence is only One reality, Brahma, and the first manifestation of Brahma expressed as Om. For this reason, Om is considered as a foundational idea and reminder, and thus is prefixed and suffixed to all Hindu prayers. Om is also considered the most sacred mantra and the most commonly used.

Om ( ॐ; in Devanagari as ओं oṁ [õː], औं auṃ [ə̃ũ], or ओ३म् om [õːm]) is a mantra and mystical sound of Hindu origin (geographically India and Nepal), sacred and important in various Dharmic religions such as Hinduism, Buddhism, Sikhism and Jainism. As to its precise graphic form, the Vedic or Indian om is what most Westerners are used to.

It is placed at the beginning of most Hindu texts as a sacred incantation to be intoned at the beginning and

end of a reading of the Vedas or prior to any prayer or mantra. It is used at the end of the invocation to the god being sacrificed to (anuvakya) as an invitation to and for the latter to partake of.

Hindus believe that as creation began, the divine, all-encompassing consciousness took the form of the first and original vibration manifesting as sound "OM".

The vibration of "OM" symbolises the manifestation of God in form ("sāguna brahman"). "OM" is the reflection of the absolute reality, it is said to be "Adi Anadi", without beginning or the end and embracing all that exists.

The mantra "OM" is the name of God, the vibration of the Supreme. When taken letter by letter, A-U-M

represents the divine energy (Shakti) united in its three elementary aspects: Bhrahma Shakti (creation), Vishnu Shakti (preservation) and Shiva Shakti (liberation, and/or destruction)

"Om" The sound of the universe. It's the first, original vibration, representing the birth, death and re-birth process.
For thousands of years, people have used this mantra to expand their awareness of the divine.
Chanting the sound OM brings us into harmonic resonance with the universe – this is a scientific fact! OM is said to vibrate at 432 Hertz, which is the natural musical pitch of the Universe, as opposed to 440 Hertz, which is the frequency of most modern music.

This is one word that anyone can feel comfortable using, no matter your spiritual inclinations. It is an incredibly soothing sound, especially when you allow the vibration to linger in your throat.
You can literally feel your body vibrate when you repeat this mantra for a little while...
For starters, you can focus on the silence from which "Om" originates and into which it dissipates. That will quickly give your mind something to focus on and help you enter and stay in meditation.

Source: Wikipedia; The hidden power in humans, Paramhans Swami Maheshwarananda, Ibera Verlag; Rituals and Mantras, Rules without meaning, Frits Staal