Sinha is a Sanskrit term originates in the Indian subcontinent, common in India and Sri Lanka. It comes from the Sanskrit word "Siṃha" ('ng' sound), which means lion.
In Sri Lanka, the term 'Sinha' (or Siha / Sinhe / Singhe / Singha) have commonly been used by the Sinhalese (or Sinhala). When it comes to the term 'Sinhala' itself, the first part of the word, 'Sinha' stands for 'lion' while 'la' or 'le' stands for 'Blood', giving the meaning 'Lions blood'. The word Simhmam (or Singam / Singham / Singhai / Singai) is the Sri Lankan Tamil derivative. In northern and middle part of India Sinhraj is also use, giving the meaning Lion king.
The more common surnames, "Singh" in India and "Weerasingha", "Jayasinghe", "Wijesinghe", "Edirisinghe", "Singaiariyan" etc. in Sri Lanka share this same root. As a surname, Sinha is mainly used by Kandyan, Kshatriya, Kurmi - Rajput's and Kayastha's as well as some Brahmins (Bhumihar), in Orissa and South West Bengal (Northern Rurh region-Burdwan belt), Sinha is also a shortened version of the surname Singha-Maha-Patra, a Utkal Brahmin surname. In Gujarat, the Sinha / Sinh surname is mostly used by Vaishnav Vaniyas, a trader class and rajputs.
Being landholding sub-castes of Brahmins, among Bhumihar Brahmins and Maithil Brahmins, some of them have secular titles like Singh interchangeably as Sinha. The general editor of the book "People of India (Bihar and Jharkhand)", published by Anthropological Survey of India (ASI), and noted academician-bureaucrat, the late Kumar Suresh Singh, said that the surname "Singh", which used to denote connection with power and authority, was used in Bihar by Brahmin zamindars, like the surname "Khan" in Muslims.
Sinha is also an unrelated name found in Brazil.
Read more about Sinha: Notable People Named Sinha, Notable Destinations Named After Sinha