I'll begin with the colleges that you can gain admission into through NEET-UG entrance examination.

1) If you land up in the merit list, you are eligible for applying in ANY of the government medical colleges across the nation BARRING colleges of Jammu and Kashmir, Andhra Pradesh and Telangana. Your admission into any of the colleges under All India 15% quota is purely merit based.

You also might be knowing that NEET offers admissions into BDS, BAMS, BHMS and BUMS courses as well.

Now the criteria for domicile is that, when you gain admission through the AI 15%quota, you are expected to submit a domicile certificate as a proof of the state from which you belong to. Admissions through all India quota are however merit based like I've mentioned, however the colleges will ask you to submit your domicile certificate as a part of document collection and hence this becomes a mandatory document.

2)If you want to contest through the 85% STATE QUOTA, domicile certificate for the state from which you belong is a PRELIMINARY REQUIREMENT. As your nativity to a particular state will be verified at the time of COUNSELING ITSELF. However, you'll be asked to submit your domicile certificate after you are alloted a college based on your merit, just like the other candidates, who gain admission through AI 15%quota. 

3)Irrespective of the state to which you belong to, you can contest for colleges like AIIMS, JIPMER for which there are separate examinations conducted every year. 

4)For private colleges like Kasturba medical college and COMEDK, you only need to appear for NEET, since these colleges are accepting NEET scores and are conducting merit based admission based on this examination. THERE IS DOMICILE CRITERIA FOR ADMISSION INTO THESE COLLEGES, AND STUDENTS FROM ANY CORNER OF THE COUNTRY CAN TAKE ADMISSION HERE. 

There are separate application form for appearing for counseling which is released after the results of NEET are declared. 

5)For government colleges like AMU, AFMC, BHU, again, domicile certificate will not assume importance and students from any part of the country can gain admission into this college based on merit. 

THE GIST IS, YOUR DOMICILE CERTIFICATE IS AN ABSOLUTE NECESSITY WHEN YOU ARE CONTESTING FOR THE 85%QUOTA IF YOUR STATE FOR THE GOVERNMENT MEDICAL COLLEGES. At other colleges across the nation, your domicile certificate is just like other important documents that serve as a proof of identity.

Domicile is not required for the admissions in private colleges all over India. I have known people who got admissions in the private colleges without any domicile. For instance, a person from Tamil Nadu may get admission in any of the private colleges in Kerala or Karnataka.