There are many National Research Institutes in India, so here I am listing the names of a few good institutes:

Central Tool Room & Training Centre
Centre for Airborne Systems
Centre for Artificial Intelligence and Robotics
Centre for Cellular and Molecular Biology
Acharya Tulsi Regional Cancer Institute and Research Centre
Advanced Numerical Research and Analysis Group
Aerial Delivery Research and Development Establishment
Aeronautical Development Establishment
Agharkar Research Institute
Agumbe Rainforest Research Station
AISSMS College of Pharmacy
Indian Institute of Plantation Management
Indian Institute of Pulses Research
Jawaharlal Nehru Centre for Advanced Scientific Research
Laser Science and Technology Centre
Liquid Propulsion Systems Centre
Indian Institute of Remote Sensing
Indian Institute of Nano Science & Technology
All India Institute of Speech and Hearing
Anna University K B Chandrashekar Research Centre
Anthropological Survey of India
ANURIB
National Institute of Oceanography, India
National Institute of Public Finance and Policy
Arid Forest Research Institute
Arignar Anna Memorial Cancer Hospital & Research Institute
Armament Research and Development Establishment
Aryabhatta Research Institute of Observational Sciences
Automotive Research Association of India
National Institute of Ayurveda
National Institute of Hydrology
Defence Institute of Advanced Technology
Defence Materials and Stores Research and Development Establishment
National Institute of Mental Health and Neurosciences
National Institute of Ocean Technology
Gau Vigyan Anusandhan Kendra
Gokhale Institute of Politics and Economics
Govind Ballabh Pant Social Science Institute
National Institute of Science Communication and Information Resources
National Institute of Science Education and Research
Electronics and Radar Development Establishment
IRADe
Environmental Protection Training and Research Institute
National Institute of Science, Technology and Development Studies
National Institute of Unani Medicine