These three exams cover a huge share of undergraduate admissions in India, and it's easy to confuse what each one is actually gatekeeping.
NEET-UG — the only route into MBBS/BDS
NEET-UG (National Eligibility cum Entrance Test), conducted by the National Testing Agency (NTA), is the sole entrance route for MBBS and BDS admission at every recognised medical and dental college in India — government or private. There is no direct-admission or alternative-exam pathway for these two degrees.
JEE Main and JEE Advanced — engineering's two-stage gate
JEE Main (also conducted by NTA) is the entrance exam for admission to NITs, IIITs, and other centrally-funded technical institutions, and it also serves as the qualifying stage for JEE Advanced. JEE Advanced, conducted by one of the IITs on a rotating basis, is required specifically for admission to the IITs. Many state and private universities also accept JEE Main scores directly or run their own separate entrance test alongside it.
CUET-UG — a common gate for central and many other universities
CUET-UG (Common University Entrance Test), also conducted by NTA, is used for undergraduate admission to Central Universities and has been adopted by a growing number of state, deemed and private universities as well — but adoption is institution-by-institution, so it is not universal the way NEET is for medical admission.
The practical takeaway
Which exam(s) you need depends entirely on the specific course and specific institutions you're targeting — use EduAspire's Entrance Exam Finder against your target course, and always confirm the current year's accepted exams directly on the institution's own admission page before assuming.