Hello Avinash,
Look there are three ways to take admission into any college. Firstly through JEE main score, second through College entrance exams and third is through management quota. Now, let's take an example of JEE main exam.
Initially, CBSE declares JEE main score and top 1,15,000 students apply for JEE Advance. As soon as results of respective boards are out CBSE asks students to update their board results by logging into their respective account on JEE main website. If you fail to update your board marks before the time your rank will be out on the basis of JEE main.Then CBSE declares JEE main Rank. Based on that rank you can take admission in any college throughout INDIA.
Every college has two to three round of counseling. Each college has a different parameter to call students like they set some cut-off. Based on that they call students. students can attend the counseling by paying the counseling fee and choose the branch of their choice.
As soon as counseling results are out and students get to know about their allotted seats and are asked to pay the admission fee by visiting the institute they've been allotted. If you didn't get the choice of your branch you still can opt-out of that college and they will refund the counseling fee. This way you get a seat in any engineering college.