What is "kaushik"

Rajarshi Koushika/Kaushik is a King of Chandravansha Clan later became as the Royal Sage and got name as Brahmarshi Vishvamitra from Sage Vashistha. Vishvamitra is also pronounced as "Vaishvamitra", as can be seen in gotras and abhivadhanam, a traditional/Vedic way of introducing one's self (for men only) while getting blessings from elders or teachers or scholars.

He was a valiant warrior and the great-grandson of a great king named Kusha. The Valmiki Ramayana, prose 51 of Bala Kanda, starts with the story of Vishvamitra:

It is mentioned in Srimad Bhagavatam 9.15.4 "The son of Jahnu was Puru, the son of Puru was Balāka, the son of Balāka was Ajaka, and the son of Ajaka was Kuśa. Kuśa had four sons, named Kuśāmbu, Tanaya, Vasu and Kuśanābha. The son of Kuśāmbu was Gādhi." Then in Srimad Bhagavatam 9.16.28 "Viśvāmitra, the son of Mahārāja Gādhi, was as powerful as the flames of fire. From the position of a kṣatriya, he achieved the position of a powerful brāhmaṇa by undergoing penances and austerities."

Kaushika or Kaushik is gotra of Brahmins mainly found in India.

There was a Rishi (saint) by the name of "Vishvamitra" literally meaning 'friend of the universe','vishwa' meaning universe and 'mitra' meaning friend. He was also called as Rishi ' Kaushika'. Vishvamitra is famous in many legendary stories and in different works of Hindu mythology.