The Reserve Bank of India is the central Bank that is fully owned by the Government of India.
It is governed by a central board (headed by a Governor) appointed by the Central Government.
It issues guidelines for the functioning of all banks operating within the country.

Please find below the list of Public, Private & Foreign Banks list.
Public Sector Banks :
1. State Bank of India
2. State Bank of Hyderabad
3. State Bank of Bikaner and Jaipur
4. State Bank of Mysore
5. State Bank of Indore
6. State Bank of Travancore
7. State Bank of Patiala
8. Andhra Bank
9. Allahabad Bank
10. Bank of India
11. Bank of Maharashtra
12. Bank of Baroda

13. Canara Bank
14. Central Bank of India
15. Corporation Bank
16. Dena Bank
17. Indian Bank
18. Industrial Development Bank of India (IDBI) Ltd
19. Indian Overseas Bank
20. Oriental Bank of Commerce
21. Punjab National Bank
22. Punjab and Sind Bank
23. Syndicate Bank
24. Union Bank of India
25. UCO Bank
26. United Bank of India
27. Vijaya Bank
Private Sector Banks :
1. HDFC Bank Ltd
2. ICICI Bank Ltd
3. Axis Bank
4. Development Credit Bank Ltd
5. ING Vysya Bank Ltd
6. Karur Vysya Bank Ltd
7. City Union Bank Ltd
8. Bank of Rajasthan Ltd
9. Catholic Syrian Bank Ltd
10. Federal Bank Ltd
11. Dhanalakshmi Bank Ltd
12. Jammu and Kashmir Bank Ltd
13. Karnataka Bank Ltd
14. Lakshmi Vilas Bank Ltd
15. South Indian Bank Ltd
16. IndusInd Bank Ltd
17. Kotak Mahindra Bank Ltd
18. Yes Bank Ltd
19. United Western Bank Ltd
20. SBI Commercial and International Bank Ltd
21. Tamilnad Mercantile Bank Ltd
Foreign Banks :
1. Citibank
2. Hongkong & Shanghai Banking Corporation Ltd
3. Standard Chartered Bank
4. Deutsche Bank
5. ABN-AMRO Bank
6. American Express Bank Ltd
7. Abu Dhabi Commercial Bank Ltd
8. Barclays Bank
9. BNP Paribas
10. DBS Bank Ltd
11. State Bank of Mauritius Ltd