Apart from the exams JEE MAIN (to gain admission into NITs/IIITs and probably CFTIs) and JEE ADVANCED (to be taken by IIT aspirants), there are some of the well known institutes that have their own entrance examinations which attract huge number of aspiring engineers. Of late, these institutes also have gained reputation for the quality education they offer. Some of the institutes, apart from BITS Pilani, are Vellore Institute of Technology(VIT), Amrita Vishwa Vidyapeetham, Cochin Institute of Science and Technology(CUSAT), SRM,etc. One of the admission routes to IISERs is JEE ADV while for IIST it is through JEE ADV only. Institutes like SASTRA use JEE MAIN as one of the admission criteria.
Some of the states conduct their own entrance examination for the admission into the state-run and some of the private engineering colleges (in those states). Though some of the state entrance examinations are known beyond that state's borders, often admission into those institutes is restricted by the domicile clause.
Andhra Pradesh has AP EAMCET (Engineering,Agriculture and Medical Entrance Examination), Telangana has TS EAMCET,West Bengal has WBJEE(West Bengal Engineering Entrance Examination), to name a few.