I am currently writing a paper for the thesis work I did last year along with my advisors. (I successfully graduated last year and currently work in a software company.) I am confused regarding what organization I should mention in the author's section. Should I mention my university name since I completed my thesis work when I was a university student or the name of the company I currently work for?
I will take a somewhat different view from @jakebeal's answer.
The primary question is: when did you work on the paper? Did you do all of the work while you were a thesis student, and then the paper has slowly made its way through the system? Or did you actively work on it as part of your new job? Did you use institutional resources from your new position to do any part of the paper?
If the answer to the latter is no, then you should only list the old institution in the actual affiliations, and then list your new job as a "present address", if you so choose. If you worked on the paper as part of your new job (beyond rote submission and minor edits), then you should list both workplaces as "active" affiliations.
There is some ambiguity here, particularly since it sounds like you have headed into a non-research industrial job.
If you were in a research-oriented position, you would probably want to put down your current affiliation as your organization, and possibly the university as a secondary affiliation (some publication venues allow multiple affiliations).
If this paper is instead from a phase of your life that you are setting behind you as you depart the research community, however, it's entirely reasonable to leave your current non-research company out of it and simply claim your old university as the relevant affiliation.