Applied through university after Exxon gave an information session at my school. I was then invited to interview on campus. This initial interview lasted about 45 minutes and consisted entirely of behavioral questions. The normal "tell me about a time when..." STAR type questions. The interviewer had a packet of questions they would read from and then record my responses, with minimal eye contact or other acknowledgement of my answers. You should come prepared with five or more situations that you can discuss for different topics like showing leadership, overcoming sudden challenges, dealing with uncooperative group members, etc. Don't memorize your stories but know what things to hit on for each one.
Within a week of my initial interview I was invited to come to the Houston office for the second round. I was flown there and treated to a dinner the night before the interview. This dinner gives you a chance to meet other candidates and new college hires. The next day we were taken to the campus, where a continental breakfast was provided and we ate alongside the interviewers as well as other ExxonMobil employees. Don't be afraid to speak up here and ask the employees questions or even try to engage in small talk.
After breakfast the interviews began. It was three interviews, back to back, 50 minutes each. Each interview was one-on-one with a different interviewer, but they were all behavioral and consisted of similar questions from the first round. Same type of STAR questions, read from a packet, with your answers being jotted down. It seemed that each interviewer was also able to ask one question of their own. After that there was a lunch and a presentation. It was a very long day, and you should be in "interview mode" for the entire time. Rest easy though, knowing that if you made it to the second round of interviews then you have a VERY good chance of getting an offer. Just be yourself and show them that you're someone they would want to work with.
I received my offer within a week from the second round.