Srila Prabhupada

Sparks are beautiful as long as they are in the fire. Similarly, we have to remain in the association of the Supreme Personality of Godhead and always engage in devotional service, for then we shall always be brilliant and illuminating. As soon as we fall from the service of the Lord, our brilliance and illumination will immediately be extinguished, or at least stopped for some time. When we living entities, who are like sparks of the original fire, the Supreme Lord, fall into a material condition, we must take the  mantra from the Supreme Personality of Godhead as it is offered by Sri Caitanya Mahaprabhu. By chanting this Hare Krsna mantra, we shall be delivered from all the difficulties of this material world.
(Srimad Bhagavatam (8.6.15))
If one takes to Caitanya Mahaprabhu's method, chanting Hare Krsna, the first installment of gain will be ceto-darpana-marjanam. [Cc. Antya 20.12] All the dirty things will be cleansed from the heart simply by chanting. Chant. There is no expenditure, and there is no loss. If one simply chants for one week, he will see how much he will progress in spiritual knowledge.
(The Science of Self·Realization)
Recently, an Indian yogi came to America to give some "private mantra." But if a mantra has any power, why should it be private? If a mantra is powerful, why should it not be publicly declared so that everyone can take advantage of it? We are saying that this Hare Krsna mahamantra can save everyone, and we are therefore distributing it publicly, freeof charge… The devotees are preaching without charge, declaring in the streets, parks, and everywhere, "Here! Here is the Hare Krsna Mahamantra.Come on, take it!"
(Path of Perfection)

Meaning of Krishna

The name Krsna means "all-attractive." God attracts everyone; that is the definition of "God." We have seen many pictures of Krsna, and we see that He attracts the cows, calves, birds, beasts, trees, plants, and even the water in Vrndavana. He is attractive to the cowherd boys, to the gopis, to Nanda Maharaja, to the Pandavas, and to all human society. Therefore if any particular name can be given to God, that name is "Krsna."
(The Science of Self-Realization)

Meaning of Rama

"Rama" means spiritual enjoyment: Those who are actually yogis truly enjoy, but how do they enjoy? Ramante yogino 'nante-their enjoyment is unlimited, that unlimited enjoyment is real happiness, and such happiness is spiritual, not material. This is the real meaning of Rama, as in the chant Hare Rama. Rama means enjoyment through spiritual life. Spiritual life is all pleasure, and Krsna is all pleasure.
(The Perfection of Yoga)
"Hare Rama" refers both to Sri Balarama and Lord Ramacandra

Meaning of Hare

When we are chanting the mahamantra, we are actually addressing God and His energy, Hara. Hara is Krsna's internal potency, Srimati Raddharani. Thus the Vaisnavas worship RadhaKrsna, Laksmi-Narayana and Sita Rama. In the beginning of the Hare Krsna mahamantra we first address the internal energy of Krsna, Hare. Thus we say, "0 Radharani! 0 Hare! 0 energy of the Lord!"
(Teachings of Lord Kapila, the Son of Devahati)
When the mind is finally trained to the point where it will think of nothing but Krsna, it will attain peace and will become very tranquil.
"The yogi whose mind is fixed on Me verily attains the highest happiness. By virtue of his identity with Brahman, he is liberated; his mind is peaceful, his passions are quieted, and he is freed from sin." (Bg. 6.27)
The mind is always concocting objects for happiness. I am always thinking, "This will make me happy," or "That will make me happy. Happiness is here. Happiness is there." In this way the mind is taking us anywhere and everywhere. It is as though we are riding on a chariot behind an unbridled horse. We have no power over where we are going but can only sit in horror and watch helplessly. As soon as the mind is engaged in the Krsna consciousness process specifically by chanting, then the wild horses of the mind will gradually come under our control.
(On the Way to Krsna)
[The Visnudutas to the Yamadutas]: Authorities who are learned scholars and sages have carefully ascertained that one should atone for the heaviest sins by  ndergoing a heavy process of atonement and one should atone for lighter sins by undergoing lighter atonement. Chanting the Hare Krsna mantra, however, vanquishes all the effects of sinful activities, regardless of whether heavy or light.
One should take shelter of Krsna Balarama, the Supreme Personality of Godhead, whose protective power is so great that it cannot be equaled in tbe material world. However powerful the reactions of one's sins, they will immediately be vanquisbed if one chants the name of Hari, Krsna, Balarama or Narayana.

(Srimad-Bhagavatam 6.2.16)

"'Whatever is achieved by meditation in Satyayuga, by the performance of yajna in Treta yuga or by the worship of Krsna's lotus feet in Dvapara yuga is also obtained in tbe age of Kali simply by chanting and glorifying Lord Kesava.'"
Visnu Purana 6.2.17, Padma Pura,"a (Uttarakhada 72.25),and Brhan-naradiya Purana 38.97 {cited: Sri Caitanyacaritamrta Madhya-lila 20.346)
The holy name "Rama" 15 equal to one thousand holy names of Visnu [Siva to Durga]: "Lord Siva addressed his wife Durga as Varanana and explained, I chant the holy name of Rama, Rama, Rama and thus enjoy this beautiful sound. This holy name of Ramacandra is equal to one thousand names of Lord Vishnu.'" Brhad-visnu-sahasra-nama-stotra, 72.335
(from Padma Purana, Uttarakhanda)

'''The pious results derived from chanting the thousand holy names of Vishnu three times can be attained by only one repetition of the holy name of Krsna.'"

(Brahmanda Purana)

Srila Sridhara Swami quotes that to do what is good and to reject what is not good is called mangala, or auspicious. To do what is good means to accept everything favorable to the discharge of devotional service, and to reject what is not good means to reject everything not favorable for discharging devotional service.In our Krsna consciousness movement, we accept this principle by rejecting four prohibited items- namely illicit sex life, intoxication, gambling and flesheating  and accepting tbe daily chanting of at least sixteen rounds of the Hare Krsna mahamantra and daily meditation three times a day by chanting the Gayatri mantra. In this way one can keep his brahminical culture and spiritual strength intact. By following these principles of devotional service strictly, chanting
twenty-four hours a day the mahamantra Hare Krsna, Hare Krsna, Krsna Krsna, Hare Hare/ Hare Rama, Hare Rama, Rama Rama, Hare Hare– one makes positive progress in spiritual life and ultimately becomes completely fit to see the Supreme Personality of Godhead face to face. Because the ultimate goal of studying or understanding the Vedic knowledge is to find Krsna, one who follows the Vedic principles as described above can from the very beginning see all the features of Lord Krsna, the Absolute Truth, very distinctly, as one can see one's own face completely reflected in a clear mirror.
(Srimad Bhagavatam 4.21.42)