origin of INDIA's name
The name India (भारत) may refer to either the region of Greater India (the Indian subcontinent), or to the contemporary Republic of India contained therein. The term is derived from the name of the Sindhu (Indus River) and has been in use in Greek since Herodotus (4th century BC). The term appears in Old English in the 9th century, and again in Modern English since the 17th century.
The Republic of India has three principal short names, in both official and popular English usage, each of which is historically significant. All three originally designated a single entity comprising all the modern nations of the Indian subcontinent. These names are India, Bharat and Hindustan. The first Article of the Constitution of India states that "India, that is Bharat, shall be a union of states." Thus, India and Bharat are equally official short names for the Republic of India, while "Hindustan" is still widely used as an alternative name when Indians speak amongst themselves. "Hindustan" is also used in historical contexts (especially British India). Indians commonly refer to their country as Bharat, Hindustan or India depending on the context and language of conversation.
The English term is from Greek Indía (Ἰνδία), via Latin India. Iindía in Byzantine (Koine Greek) ethnography denotes the region beyond the Indus (Ἰνδός) river, since Herodotus (5th century BC) ἡ Ἰνδική χώρη "Indian land", Ἰνδός "an Indian", from Avestan Hinduš (referring to Sindh, and listed as a conquered territory by Darius I in the Persepolis terrace inscription). The name is derived ultimately from Sindhu, the Sanskrit name of the river, but also meaning "river" generically. Latin India is used by Lucian (2nd century).
The name India was known in Old English, and was used in King Alfred's translation of Orosius. In Middle English, the name was, under French influence, replaced by Ynde or Inde, which entered Early Modern English as Indie. The name India then came back to English usage from the 17th century onwards, and may be due to the influence of Latin, or Spanish or Portuguese. 
Sanskrit indu "drop (of Soma)", also a term for the Moon, is unrelated, but has sometimes been erroneously connected, listed by, among others, Colonel James Todd in his Annals of Rajputana. Todd describes ancient India as under control of tribes claiming descent from the Moon, or "Indu", (referring to Chandravanshi Rajputs), and their influence in Trans-Indian regions where they referred to the land as Industhan.
The name  is used for the Republic of India. Bhārata is the official Sanskrit name of the country, Bhārata Gaṇarājya, and the name is derived from the ancient Indian texts, the Puranas, which refers to the land that comprises India as Bharata varṣam, and uses this term to distinguish it from other varṣas or continents. For example, the Vayu Purana says he who conquers the whole of Bharata-varsa is celebrated as a samrāt (Vayu Purana 45, 86).. However in some puranas, the term 'Bharate' refers to the whole Earth as Emperor Bharat had ruled the whole Earth, due to the limited knowledge of global geography at the time. Until the death of Maharaja Parikshit, the last formidable emperor of the Kuru dynasty (there were other emperors too after him but they were not as powerful as him), the known world was known as Bharat varsha.
The Sanskrit word bhārata is a vrddhi derivation of bharata, which was originally an epithet of Agni. The term is a verbal noun of the Sanskrit root bhr-, "to bear / to carry", with a literal meaning of "to be maintained" (of fire). The root bhr is cognate with the English verb to bear and Latin ferō. This term also means "one who is engaged in search for knowledge".
The Bhāratas were Indians mentioned in the Rigveda, notably participating in the Battle of the Ten Kings.
The realm of Bharata is known as Bharātavarṣa in the Mahabhārata (the core portion of which is itself known as Bhārata) and later texts. The term varsa means a division of the earth, or a continent.  A version of the Bagavatha Purana says, the Name Bharatha is after Jata Bharatha who appears in the fifth canto of the Bagavatha.
From: Vishnu Purana (2.3.1)
uttaraṃ yatsamudrasya himādreścaiva dakṣiṇam
varṣaṃ tadbhārataṃ nāma bhāratī yatra santatiḥ
उत्तरं यत्समुद्रस्य हिमाद्रेश्चैव दक्षिणम् ।
वर्षं तद् भारतं नाम भारती यत्र संततिः ।।
"The country (varṣam) that lies north of the ocean and south of the snowy mountains is called Bhāratam; there dwell the descendants of Bharata."
The term in Classical Sanskrit literature is taken to comprise the territory of Republic of India, Pakistan, Nepal and Bangladesh, as well as portions of eastern Afghanistan. This corresponds to the approximate extent of the historical Maurya Empire under emperors Chandragupta Maurya and Ashoka the Great (4th to 3rd centuries BC). Later political entities unifying approximately the same region are the Mughal Empire (17th century), the Maratha Empire (18th century) , and the British Raj (19th to 20th centuries).
("undivided Bharat") is a term of Hindu nationalism calling for a re-unification of the region.
From the perspective of the Malayans, Indian traders were the most common ones coming from "the West", therefore the word was absorbed into the Malay language. In the Malay language, "Barat" literally means "West".
For Middle Eastern traders, particularly Arabs and Turks, spices were the most common materials coming from the East. Hence, the term "bharat" was borrowed first into Arabic as بهرات baharāt, meaning "spices;" this migrated into the Turkish "baharat" with the same meaning.
Hindustan and Hind
The name Hind is derived from the Iranian equivalent of Indo-Aryan Sindh. The Avestan -stān means country or land (cognate to Sanskrit sthāna "place, land").
India was called Hindustan in Persian although the term Hind is in current use. al-Hind الهند is the term in the Arabic language (e.g. in the 11th century Tarikh Al-Hind "history of India"). It also occurs intermittently in usage within India, such as in the phrase Jai Hind.
The terms Hind and Hindustān were current in Persian and Arabic from the 11th century Islamic conquests: the rulers in the Sultanate and Mughal periods called their Indian dominion, centred around Delhi, Hindustan. -stan is a Persian suffix meaning "home of/place of".
Hindustān, as is the term Hindu itself, entered the English language in the 17th century. In the 19th century, the term as used in English referred to the northern region of India between the Indus and Brahmaputra rivers and between the Himalayas and the Vindhyas in particular, hence the term Hindustani for the Hindi-Urdu language. Hindustan was in use synonymously with India during the British Raj.
Hind (Hindi: हिन्द) remains in use in Hindi-Urdu. In contemporary Persian language, the term Hindustan has come to mean the Indian subcontinent, and the modern Indian Union is called Hind. The same is the case with Arabic language, where al-Hind is the name of the Republic of India.
Tenjiku is the Japanese word commonly used in reference to pre-modern India. Tian, the root word for the Japanese kanji means, 'heaven,' while, jiku, means: 'the center of', or 'primary concentration of'. Therefore the word is known to mean: 'the heavenly center of the world" or "the spiritual axis (center)", a reference to the Indian origins of Buddhism.
There is a story in Jain mythology and in Hindu and Buddhist texts as well that describe Jambudvipa being one of the seven islands/continents of the world. It is possible that perhaps "island" is used to refer to India because India in one time was a separate Indian Plate. Perhaps the phrase is used in the manner that the land of India is still an island in between the Indus River and the rest of the Asian Plate. Jambudvipa is also addressed in Buddhist mythology and in some he is addresses as the ruler over entire Jambudvipa and in one, "He reigned over a quarter of the land of Jambudvipa..."
Some historical definitions
Some historical definitions prior to 1500
c. 486 BC Hidush Naksh-i-Rustam
"Says Darius the King: By the grace of Ormazd these (are) the countries which I have acquired besides Persia. I have established my power over them. They have brought tribute to me. That which has been said to them by me they have done. They have obeyed my law. Medea... Arachotia (Harauvatish), Sattagydia (Thatagush), Gandaria (Gadára), India (Hidush)...."
c. 440 BC India Herodotus
"Eastward of India lies a tract which is entirely sand. Indeed, of all the inhabitants of Asia, concerning whom anything is known, the Indians dwell nearest to the east, and the rising of the Sun."
c. 300 BC India/Indikē Megasthenes
"India then being four-sided in plan, the side which looks to the Orient and that to the South, the Great Sea compasseth; that towards the Arctic is divided by the mountain chain of Hēmōdus from Scythia, inhabited by that tribe of Scythians who are called Sakai; and on the fourth side, turned towards the West, the Indus marks the boundary, the biggest or nearly so of all rivers after the Nile."
c. 140. Indoi, Indou Arrian
"The boundary of the land of India towards the north is Mount Taurus. It is not still called Taurus in this land; but Taurus begins from the sea over against Pamphylia and Lycia and Cilicia; and reaches as far as the Eastern Ocean, running right across Asia. But the mountain has different names in different places; in one, Parapamisus, in another Hemodus; elsewhere it is called Imaon, and perhaps has all sorts of other names; but the Macedonians who fought with Alexander called it Caucasus; another Caucasus, that is, not the Scythian; so that the story ran that Alexander came even to the far side of the Caucasus. The western part of India is bounded by the river Indus right down to the ocean, where the river runs out by two mouths, not joined together as are the five mouths of the Ister; but like those of the Nile, by which the Egyptian delta is formed; thus also the Indian delta is formed by the river Indus, not less than the Egyptian; and this in the Indian tongue is called Pattala. Towards the south this ocean bounds the land of India, and eastward the sea itself is the boundary. The southern part near Pattala and the mouths of the Indus were surveyed by Alexander and Macedonians, and many Greeks; as for the eastern part, Alexander did not traverse this beyond the river Hyphasis. A few historians have described the parts which are this side of the Ganges and where are the mouths of the Ganges and the city of Palimbothra, the greatest Indian city on the Ganges. (...) The Indian rivers are greater than any others in Asia; greatest are the Ganges and the Indus, whence the land gets its name; each of these is greater than the Nile of Egypt and the Scythian Ister, even were these put together; my own idea is that even the Acesines is greater than the Ister and the Nile, where the Acesines having taken in the Hydaspes, Hydraotes, and Hyphasis, runs into the Indus, so that its breadth there becomes thirty stades. Possibly also other greater rivers run through the land of India."
c. 590. Hind Istakhri "As for the land of the Hind it is bounded on the East by the Persian Sea (i.e. the Indian Ocean), on the W. and S. by the countries of Islām, and on the N. by the Chinese Empire. . . . The length of the land of the Hind from the government of Mokrān, the country of Mansūra and Bodha and the rest of Sind, till thou comest to Kannūj and thence passest on to Tibet, is about 4 months, and its breadth from the Indian Ocean to the country of Kannūj about three months."
c. 650 Five Indies Xuanzang
"The circumference of the Five Indies is about 90,000 li; on three sides it is bounded by a great sea; on the north it is backed by snowy mountains. It is wide at the north and narrow at the south; its figure is that of a half-moon."
c. 944. Hind, Sind Masudi "For the nonce let us confine ourselves to summary notices concerning the kings of Sind and Hind. The language of Sind is different from that of Hind. . . ."
c. 1020 Hind Al-Birūnī
"Hind is surrounded on the East by Chín and Máchín, on the West by Sind and Kábul, and on the South by the Sea."-
1205 Hind Hasan Nizāmī "The whole country of Hind, from Peshawar in the north, to the Indian Ocean in the south; from Sehwan (on the west bank of the Indus) to the mountains on the east dividing from China."
1298 India the Greater
India the Minor
Middle India Marco Polo
"India the Greater is that which extends from Maabar to Kesmacoran (i.e. from Coromandel to Mekran), and it contains 13 great kingdoms. . . . India the Lesser extends from the Province of Champa to Mutfili (i.e. from Cochin-China to the Kistna Delta), and contains 8 great Kingdoms. . . . Abash (Abyssinia) is a very great province, and you must know that it constitutes the Middle India."
c. 1328. India Friar Jordanus "What shall I say? The great- ness of this India is beyond description. But let this much suffice concerning India the Greater and the Less. Of India Tertia I will say this, that I have not indeed seen its many marvels, not having been there. . . ."
1404 India Minor Clavijo "And this same Thursday that the said Ambassadors arrived at this great River (the Oxus) they crossed to the other side. And the same day . . . came in the evening to a great city which is called Tenmit (Termez), and this used to belong to India Minor, but now belongs to the empire of Samarkand, having been conquered by Tamurbec."
Republic of India
The official names as set down in article 1 of the Indian constitution are:
English: India; Bharat
Hindi: भारत (Bhārat)
Source:From Wikipedia, the free encyclopedia