There are 49 established Lead Hindu Gothras (or Gothram). All members of a particular Gothra are believed to possess certain common characteristics by way of nature or profession.
The term Gothra was used in its present sense for the first time in the Brahmanas. It was systematised by about the 4th century BC to accommodate changed social rules and laws and by the time of the Sutras, it was a well-established system.
Gothras have their orgination to saptharshies who change with Manvanthara. We are in the seventh Manvanthara now.
Many of the seven sages have been repeated and replaced. In the first manvanthara the saptharshies were Marichi, Atri, Angeerasa, Pulasthia,Pulaha Kratu  and Vasistha. They are believed to be the mind-born sons of Brahma.
According to the Baudhâyanas'rauta-sûtra Vishvâmitra, Jamadagni, Bharadvâja, Gautama, Atri, Vasishtha, Kashyapa and Agastya are 8 sages; the progeny of these eight sages is declared to be Gothras.
Section   I) 1. Gouthama Gothra 2. Garga Gothra 3. Agasthia Gothra 4. Bhargava Gothra 5. Bharadwaja  & 6. Atri Gothra.
Section  II1. Haritha Gothra. 2. viswamithra Gothra 3. Vasistha Gothra 4. Vadula Gothra 5. Upamanyu Gothra 6. Shounaka Gothra  & 7. Sankrithi Gothra. 
Section III) 1. Moudgalya Gothra 2. Sandilya Gothra 3. Salakhyana Gothra  4. Raivata Gothra 5. Koundinya Gothra 6. Mandaya Gothra 7. Maitreya Gothra & 8. Katayana Gothra.
Section IV) 1. Dhanwantari Gothra 2. Jamadagni Gothra 3. Kanva Gothra  & 4. Kātyāyana Gothra.
 Section I
1. Gouthama Gothra 2. Garga Gothra 3. Agasthia Gothra 4. Bhargava Gothra 5. Bharadwaja  & 6. Atri Gothra.
1. Gouthama Gothra.
Gautama Maharishi is one of the Saptarishis of the current Manvantara (seventh). He was one of the Maharishis of Vedic times, known to have been the discoverer of Mantras -- 'Mantra-drashtaa', in Sanskrit
The Rig Veda has several suktas that go with his name. He was the son of Rahugana, belonging to the line of Angiras. The Devi Bhagavatam says that the river Godavari is so named because of its association with Gautama. He had two sons by name Vamadeva and Nodhas, both themselves discoverers of Mantras
There is a hymn called Bhadra in the Sama Veda which again is ascribed to Gautama Maharishi. His wife is Ahalya,
The Puranas speak of the story wherein it is described how Gautama won the hand of Ahalya by perambulating the divine cow.
The Chief priest of King Janaka of Mithila, by name Shatananda, was the son of Gautama and Ahalya. Gautama's sixty-year long penance is mentioned in the Shanti parva of the Mahabharata
The Narada purana describes the story of the 12-year famine during which Gautama fed all the Rishis and saved them.
The Brahmaanda-purana mentions that this Gautama initiated one of the sub-branches of the Raanaayani branch of Sama Veda.
Some famous disciples of Gautama were Praachina-yogya, Shaandilya, Gaargya, and Bharadwaja.
According to the Ramayana, Rishi Gautama once went to take bath in the river Ganges early morning. The king of the devas, Indra, was fascinated with Gautam's wife, Ahalya. Indra came in the form of Gautam and made love to Ahalya ….
Gauatama was also the author of Dharma-sutra known as Gautama Dharma sutra [2] [3]. It is in fact the earliest Dharma Sutra
Sage  Gautama was the most ancient sage of all Brahmin lawgivers. He was quoted by Baudhayana and belonged to Samaveda School.  Gautama’s teachings are called Gautamasutra or Gautamasmriti.
Gautamas Brahmins are originally settled in Brij region of North India
2. Garga Gothra
Garga is the son of Rishi Bharadwaja and Suseela .  Gargya (son of Garga) is  the author of some of the Sukthas of the Atharvana Veda. Sage garga was the family priest of the family of Nanda (the foster-father of Krishna). He named child  as "Krishna" after receiving the name by meditation. Garga is the author of Garga Samhita.
The Brihat Parasara Hora Sastra In this sastra in chapter 21, Parasara quotes Garga and Brahma on the effects of the 10th bhava (house).
Sri Prasanna Parvathi Sametha Gargeshwari temple near Mysore is named after the Garga where Ardhanareeswara appeared before him.
3. Agsthaia Gothram
Agastya  was a Vedic sage. Agastya and his clan are also credited to have "authored" many mantras of the Rig Veda Agastya is also the author of Agastya Samhita In some reckonings, Agastya is the greatest of the Seven Sages or Saptarshis. The word is also written as Agasti. A-ga means a mountain, Asti, thrower
Agastya the Rishi, was born of Gods Varuna, from Urvashi .Another reference to him  is in the Mahabharata in Sauptikaparva as the teacher of Guru Drona.
As with all other Hindus, it was necessary for Agastya to marry and sire a son, in order to fulfill his duties to the Manus. Once he resolved upon doing this, Agastya pursued an unusual course of action.
By his yogic powers, he created a female infant who possessed all the special qualities of character and personality that would be appropriate in the wife of a renunciate. At this time, the noble and virtuous king of Vidarbha was childless and was undergoing penances and prayers for the gift of a child. Agastya arranged for the child he had created to be born the daughter of that noble king of Vidarbha.
The child was named "Lopamudra" by her parents. Agastya approached the king and sought the hand of his daughter when she was grown up. She was utterly intent upon exchanging the palace of her father the king for the forest-hermitage of Agastya. Lopamudra and Agastya were duly married and lived a life of extraordinary felicityIt is believed that they had two sons - Bringi & Achuthan. In Mahabharata (Vana Parva: Tirtha-yatra Parva), there is mention of his penance at Gangadwara (Haridwar), with the help of his wife, Lopamudra (the princess of Vidharba) [1].
Agastya is famous for being the first siddhar in the siddhars tradition. He created many medicines, and jadhakam( Agasthia nadi?), mandhrikam and he said all of them.
Two of his students and disciples were Therayar and Tholkappiar.
According to Akilattirattu Ammanai, the religious book of Ayyavazhi, Agastya was created from the mind of lord Siva in order to offer boons to Kaliyan (See:Boons offered to Kaliyan). As per the order of Siva, Agastya offered many boons including all worldly knowledge to him.
Sage Agastya appeared to Rama when he was despondent at the impending war with Ravana and instructed him in the use of Aditya Hridayam, a hymn praising the Sun God. Agastya also composed Saraswati Stotram.
4. Bhargava Gothram
Maharishi Bhrigu was one of the seven great sages, one of the Saptarshis in ancient India, one of many Prajapatis (the facilitators of Creation) created by Brahma (The God of Creation), the first compiler of predictive astrology, and also the author of Bhrigu Samhita, the astrological (Jyotish) classic written during the Vedic period, Treta yuga, most probably around 3000 BC.
Bhrigu is a ManasaPutra (wish-born-son) of Lord Brahma, who simply wished him into existence, to assist in the process of creation, for this reason he is also considered one of the Prajapatis.
He is married to Khyati, the daughter of Daksha. He has two sons by her, named Dhata and Vidhata.
He had one more son, who is better known than Bhrigu himself - Shukra. The sage Chyavana  coming in the pravara of Srivatsa Gothra is also his son.
Sage Bhrigu finds mention in the Vayu Purana, where he shown present during the great Yagna of Daksha Prajapati (his father-in-law).
The Bhrigus, also known as Bhargavas, are a clan of sages descending from the ancient fire-priest Bhrigu. They instituted the ritual of offering the juice of the Soma plant to the old deities This treatise is said to contain over 5 million horoscopes, in which he wrote down the fate of every being in the universe.
Bhrgu lineage: Bhrgu was the son of Brahma. (Mahabharata, Pauloma Parva)
Bhrgu descendants: Bhargavas: Bhrugu was the father of Sukracharya, the grandfather of Devayani and the great-grandfather of Yayati and the great-great-grandfather of Yadu. Indra’s daughter Jayanti was married to Bhrgu’s son Sukracharya (who also called Kavya). (Devi Bhagavatham).
Bhrgu was the grandfather Rchka (Richika), great-grandfather of Jamadagni, great-great-grandfather of Parasurama. Rchka was the son of Cyavana. (Mahabharata).
Cyavana was the son of Bhrgu’ through  wife Pauloma, who married Sukanya.
Bhrgu and Bharadwaja had discussions on many subjects. (Mahabharata).

5. Bharadwaja Gothra

The Marut Devatas found sage Bharadwaja near ganga river, raised him and taught him about the Vedas. He was adopted by Bharata, the son of Sakuntala and Dushyanta.
He performed a yajna so that his foster father Bharata would have another son (Bhumanyu) and handed that kingdom back to him.
He was a disciple of Gauthama Maharshi as well as of Valmiki. He was a first hand witness to the incident of the Krauncha birds.
He married Suseela and had a son called Garga. His son Dronacharya was born as a result of his attraction to an Apsara Ghrtaci. He trained Drona in use of weapons. Drona also learnt the use of weapons from Agnivesha, Parasurama’s student and from Parasurama himself.
Bharadwaja had a daughter called Devavarnini. She was given in marriage to Visravas and was the mother of Kubera.
Yajnavalkya, the author of the Satapatha Brahmana was a descendant of Bharadwaja.
Bharadwaja was a host to Dasaratha’s son Bharata when he was en route to meeting Sri Rama, to persuade him to return to Ayodhya.
Bharadwaja had a debate with Bhrugu about the caste system and he said that physiologically there was no difference between members of any caste. He performed the Putrakameshti yajnam for Divodasa, so that he could get a son.
Bharadwaja’s Vedic mantras were placed in the sixth Mandala of the Rig Veda by Veda Vyasa.
Dharmasutra and Srautasutra were written by Bharadwaja. The manuscript of the latter was in Pandu script and is available with the Visvavidyalaya of Bombay(Mumbai).
As per the Rktantra, pratisakhya of the samaveda, Brahma taught grammar to Brhaspati who taught it to Indra, who in turn taught it to Bharadwaja.
He was one of the great sages (rishis) descendant of rishi Angirasa, whose accomplishments are detailed in the Puranas.
6. Atri Gothra. 
In Hinduism, Attri is a legendary bard and scholar, and a son of Brahma, and one of the Saptarishis in the seventh, i.e the present Manvantara .
Attri is also a  rishi present in all manvantras. He was among the three main seers who propounded the sacred thread (after Brihaspati) which has three strands symbolising Creation (Brahma and the letter A), sustenance (Vishnu and the letter U) and Dissolution (Shiva{m} and the letter M).
Atri Gothra is from the lineage of Brahmarsi Atri and Anusuya Devi. Brahmarsi Atri is the seer of the fifth mandala (book) of the Rigveda. He had many sons, including Soma, Datta, and Durvasa.
Atri's wife is Anasuya or Anusiya devi, a daughter of Kardama Prajapati and an embodiment of chastity.
Rama, the son of Dasaratha, visited Atri Maharishi's Ashram during his fourteen years of stay in the forest. It was Atri who showed the way to Dandakaranya forest to Rama, after showering his hospitality on him.
There were also other great Rishis in that line: Mudgala, Uddaalaki, Shaakalaayani, Chaandogya, etc.Attri-samhita and Attri-smriti are two works attributed to Attri.  
a. Haritha Gothra. b. viswamithra Gothra c.Vasistha Gothra d.Vadula Gothra e. Upamanyu Gothra f.shounaka Gothra g. sankrithi Gothra.