The following explanations on the Mantras used in Meditation for Avatars (except for the Buddhist Mantra OM Mane Padme Hum) are quotations of the authorative text "Meditation and Mantras" by Swami Vishnu-devananda from 1978, published by OM Lotus Publishing Company, the international Sivananda Yoga Vedanta Centers.
Nirguna Mantras (Abstract Mantras)
Nirguna Mantars are without form. There are no deities or personalized aspects of God to be invoked. Rather, one uses the abstract Mantras and vedantic formulas to assert identification with all of creation. Because people are of many different temperaments, not all spiritual aspirants are drawn to a personal deity. Many perceive the universe as diverse energy patterns, all connected and interrelated, and stenning from one source or primal cause. For this type of temperament, the abstract Mantra creates a vibration in which the meditator identifies with the whole of the cosmos. With the repetition of one of these Mantras, the meditator loses his individual identity and merges with Nature. He avows that he is identical with that homogeneous substratum, that energy or power of existence, which underlies and permeates all that exists.
I. Soham / I am That I am 
The meditator is existence itself. He is without form, without quality, without past, present or future. No bonds or limitations restrict the aspirant who has Soham firmly fixed in his mind.
II. Aham Brahma Asi / I am Brahman
Aham Brama Asi is a great Vedantic formula. The meditator asserts himself to be One with the ever-present Brahman (wikisource) . In doing so, he denies confinement to the body and mind, and affirms unity with the Absolute.
III. Tat Twam Asi / That Thou Art
„That“ is the eternal Brahman, and „thou“ is the meditator. Tat Twam Asi, one of the greatest Vedantic statements, identifies the individual as one and the same with Brahman, the Absolute Substratum of Creation.
IV. OM
There is no translation of OM. It consists of three letters: A,U and M. It signifies the three periods of time, the three states of consciousness and all of existence. A ist the waking state, U ist he dreaming state, and M ist he deep sleep state. OM contains nada and bindu. Nada ist he prolonged vowel sound and bindu is the humming sound, made with closed lips, with which the Mantra ends. All Mantras are hidden in OM, which is the abstract, highest Mantra of the Cosmos.
Saguna Mantras (Deity Mantras)
Mantras used by spiritual aspirants to achieve God-Realisation are called deity Mantras. They are saguna, with qualities or form-producing, and aid the conceptualization process, just as do visual symbols. In time, recitation gives rise to the actual form of the particular deity. Westerners are prone to think that the various Mantras refer to different gods, and that there is a wide diversity in the culminating experience. It must never be forgotten that the deities are aspects of the one Divinity whose grandeur is too vast for the mind to comprehend at the beginning of spiritual practice.
V. OM Nama Sivaya / Prostrations to Lord Siva
Siva (wikisource) is the lord of ascetics and recluses. He is part of the Hindu Trinity. Brahma and Vishnu, the other two parts, are associated with creation and preservation, respectively. Siva, the Cosmic Dancer, presides over the destructive energies which break up the universe at the end of each age. This is the process of the old making way for the new. In a more personal sense, it is Siva’s energy by which one’s lower nature is destroyed, making way for positve growth.
VI. Om Namo Narayanaya / Prostrations to Lord Vishnu
Narayanaya is a name of Vishnu (wikisource), the preserver of the world. After the Creation, it is the energy of Vishnu which maintains order to the universe. It is Vishnu who regularly takes a human form and incarnates on earth to benefit mankind. People who are closely involved in the running of the world and maintaining the harmony of life are drawn to this aspect of God.
VII. Om Namo Bhagavate Vasudevaya / Prostrations to the Lord God, Vasudeva
Bhagavan means Lord, referring to Vishnu. Vasudeva, meaning „He who rides in all things and whom all things abide, “ is a name of Krishna. Krishna (wikisource) is one of the most loved deities. He is considered to be world teacher for he ist the source of the Bhagavad Gita, one of the most popular eastern religious texts. People are drawn to Krishna because of his playfulness and joyful nature.
VIII. Hari OM / OM Vishnu
Hari is another name for Vishnu. It is that aspect which forgives the past actions of those who take refuge in him and destroys their negative deeds. Thus Hari is a redeemer and guide to personal salvation as well as the world Preserver.
IX. OM Sri Durgai Namah / Prostrations to Mother Durga
Supreme Divinity is without qualities or attributes, and as such it contains all qualities and attributes. The masculine principles are important, yet they must be balanced with the feminine principles. Masculine and feminine are but obverse and reverse of the same coin. Durga (wikisource) represents the motherhood aspect of God. She ist he force, or shakti, through which Divinity manifests. Durga is power. She ist the protector and benefactor. According to Hindu mythology, the chaitanya, or pure consciousness, of Brahma, Vishnu and Siva were united to form the being of Mother Durga. She is commonly pictured riding a tiger and having eight arms with which she carries flowers and weapons of protection and gives the gesture of blessing.
X. OM Sri Maha Lakshmyai Namah / Prostrations to the great Mother Lakshmi
Lakshmi (wikisource) is the bountiful provider. As Vishnu’s consort, she aids the preservation of the three worlds by bestowing wealth and abundance of a material and spiritual nature. She is pictured as a beautiful woman standing on a lotus blossom with her arms open and giving.
XI. OM Aim Saraswatyai Namah / Prostrations to Mother Saraswati
Aim is the bija of Saraswati (wikisource) , the source of all learning and knowledge of arts and music. She is Brahma’s consort and is involved with the creation of new ideas and things. Responsible for bestowing wisdom and knowledge, she is often worshipped by people in creative arts.
XII. Om Sri Maha Kalikayai Namah / Prostrations to Mother Kali
Kali (wikisource) is the divine aspect which is responsible for the destruction and eradication of negative qualities in the world. She is the transformative power of Divinity which dissolves the individual into cosmic union. Maha Kali is one of the most fearsome of all expressions of Divinity. Because of the intensity of her purgative nature, very few people are initiated into this Mantra.
Maha Mantra
XIII. Hare Rama Hare Rama, Rama Rama Hare Hare; Hare Krishna Hare Krishna,
Krishna Krishna Hare Hare / My Lord, Rama! My Lord, Krishna!
Hare is a glorified form of address for calling upon God. Rama and Krishna were two of the best known and most beloved incarnations of Vishnu. They took human birth on this earth to lead mankind to eternal salvation. This is the Maha Mantra, the easiest way for attaining God-Realisation in this present age.
Gayatri Mantra
XIV. Om Bhur Bhuvah Swah, Tat Savitur Varenyam Bhargo Devasya Dheemahi, Dhiyo Yo Nah Prachodayat /
We meditate on that Ishwara’s glory, who has created the universe, who is fit to be worshipped, who is the embodiment
of Knowledge and Light, which is the remover of all sins and ignorance. May he enlighten our intellects.
The Gayatri Mantra (wikisource) is the supreme Mantra of the Vedas. It is the one Mantra that can be commonly prescribed for all, for Gayatri is the Mother of the universe, Shakti, herself, and there is nothing She cannot do. Her Mantra purifies the mind; destroys pain, sin and ignorance; brings liberation; and bestows health, beauty, strength, vitality, power, intelligence and magnetic aura.
Buddhist Mantra
XV. Om mani padme hum / Om O Jewel Lotus One Hum
Om mani padme hum (wikisource) is probably the most famous Mantra in Buddhism, the six syllabled Mantra of the bodhisattva of compassion, Avalokiteshvara. The Dalai Lama is said to be an incarnation of Avalokiteshvara, so the Mantra is especially revered by his devotees. Read the 14th Dalai Lama‘s definition as found on Wikipedia: „It is very good to recite the mantra Om mani padme hum, but while you are doing it, you should be thinking of its meaning, for the meaning of the six syllables is great and vast... The first, Om [...] symbolize the practitioner‘s impure body, speech, and mind; they also symbolize the pure exalted body, speech, and mind of a Buddha[...]“ The path is indicated by the next four syllables. Mani, meaning jewel, symbolizes the factors of method - the altruistic intention to become enlightened, compassion, and love.[...] The two syllables, padme, meaning lotus, symbolize wisdom. [...] Purity must be achieved by an indivisible unity of method and wisdom, symbolized by the final syllable hum, which indicates indivisibility [...]. Thus the six syllables, om mani padme hum, mean that in dependence on the practice of a path which is an indivisible union of method and wisdom, you can transform your impure body, speech, and mind into the pure exalted body, speech, and mind of a Buddha [...]“