The evolution of State Bank of India can be traced back to the first decade of the 19th century. It began with the establishment of the Bank of Calcutta in Calcutta, in 1806. The bank was redesigned as the Bank of Bengal, three years later. It was the first ever joint-stock bank of the British India, established under the sponsorship of the Government of Bengal. Subsequently, the Bank of Bombay (established in 1840) and the Bank of Madras (established in 1843) followed the Bank of Bengal. These three banks dominated the modern banking scenario in India, until when they were amalgamated to form the Imperial Bank of India, on 27 January 1921, with combined role of a commercial bank and a central bank.

However, with the establishment of the Reserve Bank of India in 1935, the Imperial Bank ceased to have a central banking function. It now became a purely commercial bank and certain business restrictions on it were removed. An important turning point in the history of State Bank of India is the launch of the first Five Year Plan of independent India, in 1951. The Plan aimed at serving the Indian economy in general and the rural sector of the country, in particular. Until the Plan, the commercial banks of the country, including the Imperial Bank of India, confined their services to the urban sector. Moreover, they were not equipped to respond to the growing needs of the economic revival taking shape in the rural areas of the country. Therefore, in order to serve the economy as a whole and rural sector in particular, the All India Rural Credit Survey Committee recommended the formation of a state-partnered and state-sponsored bank. The committee proposed the take over of the Imperial Bank of India, and integrating with it, the former state-owned or state-associate banks.

Subsequently, an Act was passed in the Parliament of India in May 1955. As a result, the State Bank of India (SBI) was established on 1 July 1955.  Later on, the State Bank of India (Subsidiary Banks) Act was passed in 1959. The Act enabled the State Bank of India to make the eight former State-associated banks as its subsidiaries, with the State Bank of Hyderabad becoming the first subsidiary of the SBI.

Reference:

http://www.mapsofindia.com/on-this-day/27-january-1921-the-imperial-bank-of-india-came-into-existence

http://www.iloveindia.com/finance/bank/nationalised-banks/state-bank-of-india.html