What counts as undergraduate research?
Is more independent-study type work considered research? Or is it necessary to be actively helping a professor with their research program?


In my experience, it's best to be actively interacting with a professor. But in most cases it's unlikely you'll be actively assisting with their research programs (they have usually have graduate students/postdocs for that). 

You could be given either a small project related to their funded work or possibly some crazy idea they're unwilling to let a grad student spend time on. Or anything in between.


  What do graduate admissions committees really consider "research", though?


If you can do any work that results in something published in a journal or conference, or even a poster, will be very valuable for graduate school admissions. It doesn't matter if that work is lab or computer-based. (Better yet if you can prove (with a publication/poster) that you can do both "wet" and "dry" lab work.)


  As an undergraduate sophomore, I'm really not in a position to do groundbreaking research. What's considered a good research experience for me?


When I was a junior I emailed every professor in the department. Two responded and I ended up doing "independent research credits" with one of them. It worked out really well.