Hi,
There are different bodies conducting admission in this course.
MBBS and BDS courses:-
At all India level, counselling is conducted by MCC.  One has to register at https://mcc.nic.in/ for admission to all india colleges providing MBBS and BDS.
At State level, you will have to register with the bodies conducting counselling at state level. Each state has its own health and medical education regulatory body.
For detailed information, refer to the link given below
https://medicine.careers360.com/articles/neet-counselling
For information about other courses except MBBS and bds, refer this link
https://medicine.careers360.com/articles/medical-courses-after-12th
Hope this helps!!