Yes, it is customary to ask for references when hiring for any full time job. In every company I've worked in, references have been absolutely required. It's part of the work of "due diligence" in hiring. The purpose is to get an independent view of your abilities and personality.
Reference check conversations for entry level jobs usually don't go into detail about your professional qualifications. They cover questions like, "How well does he work as a member of a team?" and "Did he usually show up for work on time?" I always used to ask, "I will be his supervisor. Do you have any advice for me?"
When they ask for references, they should spell out who the references should be from. They may ask for three references to include one direct supervisor.
You can satisfy the supervisor reference request by naming a professor (maybe your university advisor) or the crew chief on some low-level job you did to earn money. For other references you can give the name of a fellow student who knows you and your work well.
It is a good idea, when asked for references, to say "I will get back to you first thing tomorrow with a list." Then call or write your references to tell them they may get a call. Then send the list.
One of the things university professors do for work is answer requests for references from students, so don't be embarrassed about asking for a professor's reference. In this case, you should write to the professor (by email) saying
Dear Dr. Whatever,
You may remember that I was in your xyz class (or whatever) in the
spring semester of 2012 (or whatever).
I happen to need a reference, because I am applying to Abc Company to
work as a Software Engineer. I believe I am qualified for this job.
I'm eager to put into practice what you and the rest of the (whatever)
department at (whatever) University taught me.
If you have time to have a conversation about me with a manager at Abc
Company, I would be grateful. That person will call or email you.
Thanks for helping me take this next step in my career.
B. S. (or whatever degree) 2013 (or whatever year)
(your telephone number)
By doing this you're respectfully asking for the reference and reminding the professor who you are.