The "Hare Rama" mantra is very famous among Hindus. Why is this mantra so important? Is it mentioned in any texts/scriptures?
The earliest mention of the Hare Krishna mantra is in the Kali Santarana Upanishad:
At the end of Dvapara-Yuga, Narada went to Brahma and addressed him thus: "O Lord, how shall I, roaming over the earth, be able to across Kali ?" To which Brahma thus replied: "Well asked. Hearken to that which all Shrutis (the Vedas) keep secret and hidden, through which one may cross the Samsara (mundane existence) of Kali. He shakes off (the evil effects of) Kali through the mere uttering of the name of the Lord Narayana, who is the primeval Purusha". Again Narada asked Brahma: "What is the name?" To which Hiranyagarbha (Brahma) replied thus:
"Hare Rama Hare Rama Rama Rama Hare Hare Hare Krishna Hare Krishna Krishna Krishna Hare Hare. These sixteen names (words) are destructive of the evil effects of Kali. No better means than this is to be seen in all the Vedas. These (sixteen names) destroy the Avarana (or the centripetal force which produces the sense of individuality) of Jiva surrounded by the sixteen Kalas (rays). Then like the sphere of the sun which shines fully after the clouds (screening it) disperse, Parabrahman (alone) shines." Narada asked: "O Lord, what are the rules to be observed with reference to it?" To which Brahma replied that there were no rules for it. Whoever in a pure or an impure state, utters these always, attains the same world of, or proximity with, or the same form of, or absorption into Brahma.
Whoever utters three and a half Crores (or thirty-five millions) times this Mantra composed of sixteen names (or words) crosses the sin of the murder of a Brahmana. He becomes purified from the sin of the theft of gold. He becomes purified from the sin of cohabitation with a woman of low caste. He is purified from the sins of wrong done to Pitris, Devas and men. Having given up all Dharmas, he becomes freed at once from all sins. He is at once released from all bondage. That he is at once released from all bondage is the Upanishad.
Now the Kali Santarana is widely considered by scholars to be a later interpolation, not an authentic Hindu scripture. (For instance, this book only says that it was composed before the sixteenth century.) This is for multiple reasons: there are no ancient commentaries of it or ancient references to it; I think it's the only Upanishad that mentions the Yuga system, which was mainly revealed to human beings in the Itihasas and Puranas; it refers to the conversation happening at the end of the Dvapara Yuga, which would only make sense if the reader is presumed to be living long after the beginning of the Kali Yuga, etc.
In any case, the Hare Krishna mantra was popularized in the sixteenth century by the Vaishnava thinker Chaitanya Mahaprabhu. Chaitanya's followers, the Gaudiya Vaishnavas, placed supreme importance on the chanting of this mantra, which is why members of ISKCON, a popular Gaudiya organization, are commonly referred to as "Hare Krishnas."
In any case, even if you don't believe in the authenticity of the Kali Santarana Upanishad, there's still nothing wrong with chanting the Hare Krishna mantra After all, as I discuss in this answer, Hindu scriptures do say that chanting the name of Krishna/Vishnu is extremely beneficial in the Kali Yuga. And the Padma Purana says that chanting the name of Rama is as good as chanting the entire Vishnu Sahasranamam. So insofar as the popularization of the Hare Krishna mantra has encouraged the utterance of Rama and Krishna's name, there's no problem with it.
(1)History of the Hare Krishna Maha Mantra:
The Hare Krishna Maha Mantra was specifically popularized by Sri Caitanya Maha Prabhu in the 16th Century and later, on His order, His followers spread it all over the world. Lord Caitanaya is considered Channa Avatar (disguised incarnation) and only after His appearance His identity became known through His own revealing it to His devotees and exploring various Vedic literature by His devotees. For some references you can see Prabhupada's commentary on Caitanya Caritamrta here and some other references here
In Srimad Bhagavatam 11.5.31 and 32 it is said:
O King, in this way people in Dvapara-yuga glorified the Lord of the universe. In Kali-yuga also people worship the Supreme Personality of Godhead by following various regulations of the revealed scriptures. Now kindly hear of this from me.
In the Age of Kali, intelligent persons perform congregational chanting to worship the incarnation of Godhead who constantly sings the names of Krsna. Although His complexion is not blackish, He is Krsna Himself. He is accompanied by His associates, servants, weapons and confidential companions.
Gaudiya Vaishnavas consider this incarnation to be Sri Caitanya Maha Prabhu and the chanting of name of Krishna to be specifically the Hare Krishna maha mantra He popularized. The importance of chanting Krishna's name in the Kali Yuga is also mentioned in Srimad Bhagavatam 12.3.51 and 52 where it is said:
My dear King, although Kali-yuga is an ocean of faults, there is still one good quality about this age: Simply by chanting the Hare Krsna maha-mantra, one can become free from material bondage and be promoted to the transcendental kingdom.
Whatever result was obtained in Satya-yuga by meditating on Visnu, in Treta-yuga by performing sacrifices, and in Dvapara-yuga by serving the Lord’s lotus feet can be obtained in Kali-yuga simply by chanting the Hare Krsna maha-mantra.
When this maha mantra was popularized by Lord Caitanya, initially Muslims didn't like it. There was also resistance from local Hindus who didn't want the mantra to be chanted by who they considered as lower class people. They lodged a complaint with the local muslim ruler, Kazi, as mentioned in Caitanya Caritamrta Adilila Chapter 17
CC Adi 17.211 — “ ‘Now the lower classes are chanting the Hare Krsna maha-mantra again and again. For this sinful activity, the entire city of Navadvipa will become deserted.
CC Adi 17.212 — “ ‘According to Hindu scripture, God’s name is the most powerful hymn. If everyone hears the chanting of the name, the potency of the hymn will be lost.
CC Adi 17.213 — “ ‘Sir, you are the ruler of this town. Whether Hindu or Muslim, everyone is under your protection. Therefore please call Nimai Pandita and make Him leave the town.’
After this Kazi passes an order to stop sankirtana. Lord Caitanya defies this order, carries on Sankirtana with hundreds and thousands of people and the whole Sankirtana party goes to Kazi's house. There the Lord debates with Kazi. Kazi realizes Lord Caitanya's identity as the Supreme Lord, becomes His devotee and passes an order that none of his successors should prevent Sankirtana. Kazi also reveals how Lord Narasimha was in his dream and threatened him not to stop Sankirtana. To this day Muslims in that area live in harmony with local Hindus. Also to note here is that you will not find any resistance to Hare Krishna mantra in general, but only to chanting it loudly and giving it freely to what some considered as lower class people. So we cannot assume that people didn't knew about this mantra prior to Maha Prabhu, they just didn't want it publicized widely. Lord Caitanya travelled extensively for six years and there was no controversy, I know of, that was documented over the authenticity of the mantra.
(2)References to the Hare Krishna Maha Mantra
Kali-santarara Upanisad of Krishna Yajur Veda states follows:
hare krishna hare krishna krishna krishna hare hare hare rama hare rama rama rama hare hare iti sodasakam namnam kali-kalmasa-nasanam natah parataropayah sarva-vedesu drsyate
The sixteen names of the Hare Krishna maha-mantra: hare krishna hare krishna krishna krishna hare hare, hare rama hare rama rama rama hare hare destroy all the inauspiciousness of the age of Kali. This is the conclusion of all the Vedas.
Other texts like Sri Ananta-samhita, Agni Purana, Padma Purana and Brahmanda Purana are also said to have references to it, which are cited in certain books of Gaudiya authors, and you can find them in these links:
(3) Importance of the Hare Krishna Maha Mantra
Now in terms of the Significance of the names Jagadananda Pandita (one of the associates of Sri Caitanya Mahaprabhu), in a book called Prema Vivarta, citing Padma Purana and Brahmanda Purana says as follows:
visnor-ekaika-namapi sarva-vedadhikam matam tadrk nama-sahasrena rama-nama-samamsmrtam
Each name of Lord Visnu is greater and more potent than the entire Vedas; and Lord Rama's name alone is superior to a thousand names of Lord Visnu. - Padma Purana
"Dear devotees of the Lord, please continuously chant, "Krsna, Krsna, Krsna, Krsna, Krsna, Krsna, he!" And also chant, "Hare Krsna, Hare Krsna, Krsna Krsna, Hare Hare; Hare Rama, Hare Rama, Rama Rama, Hare Hare'. This mantra consisting of sixteen names is perfect in every respect, and chanting this mantra gives one the highest benefit and perfection in spiritual life
sahasra-namnam punyanam triravrttya tu yat phalam eka vrttya tu krsnasya namaikam tat prayacchati
The spiritual result one obtains from chanting a thousand names of Visnu three times is attained by chanting Lord Krsna's holy name once. - Brahmanda Purana
The citations above can be found in the ebook here on page 70. (http://ebooks.iskcondesiretree.info/pdf/Gaudiya_Books%20/Jagadananda_Pandita/Jagadananda_Pandita_Prema_Vivarta.pdf)