How can an undergraduate student conducting mathematical research with a professor expect to contribute? I am currently an undergraduate student majoring in Applied Math. I have been reading recent papers published by two of my professors who I admire and I'm wondering what it would be like to conduct research with them. In what ways can an undergraduate student contribute to mathematical research? In what ways could I prepare myself to conduct mathematical research? 

To be more specific, my academic background is in History and Social Sciences which I studied intensely from High School through my first two years in college. I conducted research at University of Arizona my junior year of high school analyzing a large set of longitudinal data on a group of high schools in NYC. 
My interest in math, specifically applied math, started when I took a Computer Science class because I was curious. Since then, I've taken calculus courses, applied combinatorics, probability courses, and I'm currently enrolled in a computational geometry course and an operations research course. The reason I'm interested in pursuing undergraduate mathematical research is because I enjoyed the experience a lot when I was studying social sciences. My concern is that I don't have a strong enough background to contribute to the research my professors currently conduct. I get good grades in my classes, but I'm concerned with my lack of background for a Junior in an applied math major. Recent publications from my professors are difficult for me to understand, so I'm questioning whether I'm ready to conduct research with them. 
I'm hoping you may be able to offer insight into whether my background is sufficient for conducting research and how I may be able to strengthen my skills. I take my school work seriously and think I have a good understanding of the material I'm taught. Thank you for your time.
It's very reasonable to wonder how you, an absolute novice, could "help" experienced experts. Some people are fond of claiming that somehow "just trying hard" or "being smart" can make this possible in some way, but those are needless (and doubtful) claims that miss the mark about what the real question could be.

That is, many or most faculty are happy to mentor beginners and introduce them to the subject(s) the faculty know and like. It need not be the case that they "need help", and, if they did, sure, recruiting novices is not the best gamble. But that's not the point! The point is that experts may enjoy (and find stimulating) introducing beginners to their subject. So you help them by allowing them to do so.

It's not at all surprising that you have trouble reading their recent papers, and, again, that's essentially irrelevant. (The internet mythology about young people becoming world experts in a few weeks or months is invidious... In particular, no, you should not expect to "read those papers and then go talk to them about them.", contrary to much advice given on-line.)

An astute expert can find interesting "starter" projects for novices. It may happen that such a project produces something interesting to the expert. Or, it may be that the expert can so well foresee the outcome that the element of surprise evaporates. But that doesn't matter. Little kids don't have to immediately play sports (or whatever) with world-class professionals. But they can be coached by such professionals, if both parties are interested. I do think that's the correct analogy, though many other people on these sites are adamant otherwise.