As you asking for the names of the private and public sector bank of India the names are as follow :

public sector bank :

There are currently 27 public sector banks in India out of which nineteen are nationalised banks and six are SBI and its associate banks, and rest two are IDBI Bank and Bharatiya Mahila Bank.

State Bank and its associate
State Bank of India
State Bank of Bikaner & Jaipur
State Bank of Hyderabad
State Bank of Patiala
State Bank of Mysore
State Bank of Travancore

Nationalised banks
Allahabad Bank
Andhra Bank
Bank of Baroda
Bank of India
Bank of Maharashtra
Canara Bank
Central Bank of India
Corporation Bank
Dena Bank
Indian Bank
Indian Overseas Bank
Oriental Bank of Commerce
Punjab & Sind Bank
Punjab National Bank
Syndicate Bank
UCO Bank
Union Bank of India
United Bank of India
Vijaya Bank

Private Sector bank
Axis Bank
Bandhan Bank
Catholic Syrian Bank
City Union Bank
Development Credit Bank
Dhanlaxmi Bank
Federal Bank
HDFC Bank
ICICI Bank
IDFC Bank
UP Agro Corporation Bank
IndusInd Bank
ING Vysya Bank
Jammu and Kashmir Bank
Karnataka Bank
Karur Vysya Bank
Kotak Mahindra Bank[
Lakshmi Vilas Bank
Nainital Bank
RBL Bank
South Indian Bank
Tamilnad Merchantile Bank
Yes Bank