There are multiple exams for MBA, national level exams, state level exams, and then university level exams.

Basic eligibility is graduation with minimum 50% for most of the exams (general category)
Some of the exams are:

CAT- For IIM's and many other top private institutions
XAT- For Xaviers mainly
NMAT- Narsee Monjee

SNAP- Symbiosis
CMAT

MAT

and you can check the complete exam list here

You can apply for CAT as the registrations have started with link to CAT Application
This link also has updates of notifications of forms, exams and other relevant information.

I am also sharing a list of top MBA institutes and from which exam they take admission in.

Top MBA institutes

Hope this helps.
Good luck.