When I read research papers I often come across many things that I'm unclear about and would like to talk over with someone. My advisor is not available to do this with me as she does not have time. I'm not sure with whom should I discuss these research papers with, in order to help me understand the papers better. I am the only student who is currently being advised by my advisor. How should I go about finding people to talk through these things with, so I can better understand the research papers I'm reading?
Your advisor is not the only person to go to, to get answers to the questions that research papers are raising for you. Talk to other researchers, in your department, or online with peers at other universities.
But it does sound like you are getting insufficient advisory support. Do you really just have the one advisor? Time to build up your supervisory team.
Talk to your advisor about what's expected of you, and what's expected of them. It sounds like you've got a mismatch between your need and their resource, and it's important to get that fixed as soon as possible.
I'd suggest that you form a reading group! (As also suggested in passing by EnergyNumbers.)
What helped me out early during my PhD was to create a series of reading groups around literatures I wanted to learn. A model I often followed was to organize a weekly meeting to read 1 book or 3-5 papers with 2-5 other students. We'd usually meet for 2 hours or so. I found other students in my cohort/program and others in the university who had similar interests. Ask around! If the papers you are reading are the kinds of things that are likely to be on your general or qualifying exams, chances are pretty good that others around you will have to be reading them as well!
There are many things not clear in your question. For example, is the paper something that your advisor has asked you to read ? Or is it just something that you're browsing for your own edification ? Did the advisor say that she cannot or will not help, or that she's busy ?
I can understand an advisor finding it difficult to spare the time to explain papers that maybe even she hasn't read. But if it's something related to your work with her, then I'd expect her to help a little more. You have to realize though that just because someone is your advisor, it doesn't mean that they know more than you about every single topic :) - in fact, part of your evolution as a student will be to get to the point where your advisor asks you for help !
But I think the general answer is as EnergyNumbers indicates: find other students in your department to discuss these papers with. That's really the best way. Also, realize that working through a difficult paper, on your own or with others, is the best way to learn new material. It's a normal part of the training process.