NIT

There are some good institutes apart from IITs, NITs and IIITs. You can check this list:
https://www.shiksha.com/b-tech/ranking/top-engineering-colleges-in-india/44-2-0-0-0. This list also includes IITs, NITs and IIITs. So, I will mention the colleges other than IITs, NITs and IIITs here:
BITS-Pilani
Indian Institute of Space Science and Technology
Anna University
IIST
DTU-Delhi
NSIT-Delhi
Jadavpur University
BIT Mesra, Ranchi
PSG College of Technology
Institute of Chemical Technology, Mumbai
PEC University of Technology
IIEST, Shibpur
TIET, Patiala
HBTU, Kanpur
College of Engineering, Pune
I am doubtful about these but some rank them higher so I will let you know (I personally believe that you should avoid these colleges)
VIT-Vellore
SRM
Apart from them some good colleges which you can go through JEE or KVPY are:
IISC
IISER
Also, check Indian Institute of Statistical Institute if you are passionate about maths.
All the best.

Following are the eligibility criteria to get admission in NIT
1. As per the revised criteria by MHRD you should have at least secured 75% marks in class 12th exam or be in the top 20 percentile in class 12th exam conducted by the respective boards and for a ST/SC category candidate eligibility criteria is 65% marks in class 12th.
2. You have to appear for JEE exam for which the criteria is depending on-a. Date of Birth of the candidate applying: Born on or after October 1, 1993(your category).b. Qualifying Examination: You should have passed 12th class or qualifying exam in 2016 or 2017 also student qualifying exam in 2018 can apply.c. Number of attempts in the Qualifying Examination: JEE Main aspirant can avail only 3 attempts for the national level undergraduate engineering entrance exam.
3. For BE/B.Tech you should have cleared 12th class examination with Mathematics and Physics as compulsory subjects along with one of the Chemistry/Language /Biology/ Technical Vocational subject whileis you are going for B.Arch/Planning you should have cleared 12th class examination with Mathematics.
The 31 NITs offer admission to about 18,000 students each year at present.