To get admisison in MBBS/BDS course, There are 3 exams
NEET - NEET is a single entrance exam for admisison to all Medical/Dental College (Except AIIMS & JIPMER) across country.

AIIMS MBBS - AIIMS MBBS exam is conducted for admission in 7 campus of AIIMS (New Delhi, Bhopal, Bhubaneshwar, Jodhpur, Patna, Raipur and Rishikesh ).

JIPMER MBBS : JIPMER MBBS is conducted for admission in 2 campus of JIPMER (Puducherry- 150 seats  and Karaikal- 50 seats)

For Engineering, there are several entrance exams across country. Every state conducts own entrance exam to fill their respective state level B.tech seats (Except few states Punjab, Haryana, MP, Rajasthan, Odisha states- these states consider JEE Main score/ 12th Marks to fill their state seats).
There are few important nation exams below:

JEE Main - It is national level level exam to get admisison in top engineering colleges like NITs, IIITs, GFTIs, few other top  governement institutes like PEC Chandigarh, IIIT Delhi, NSIT Delhi, DTU Delhi  and some top private college/universities like Thapar University, Patiala, SASTRA University, Shiv Nadar Univeristy, JAYPEE Noida, LNMIIT Jaipur.
Also, JEE Main is the entry exam for JEE Advanced.
JEE Advanced : This exam is the gatway for admission in IIT/ISM.Top 2,24,000 candidates from JEE Main would be eligible to take JEE Advanced for IIT admission.
BITSAT - .BITS Pilani conducts BITSAT for admission in 3 BITS campuses (Pilani, Hyderabad and Goa).

VITEEE - Exam for 4 VIT Campuses (Vellore, Chennai, Amaravati and Bhopal)
MUOET - Exam for 3 Manipal Institutes (Manipal, Jaipur and Sikkim Manipal).
SRMJEEE - Exam for 5 campuses of SRM (Kattankulathar, Ramapuram, Vadapalani, NCR  Delhi & Amaravati).

AEEE - Exam for 4 Amrita campuses (Amritapuri, Bengaluru, Coimbatore and Chennai).

Apart from these, You can appear for COMEDK UGET, IPU CET, KVPY Etc.