There can be awkward moments when you are trying to build your professional network. It's hard to know what to say to start trying to build a professional relationship with someone. I've found though, that there is one simple question that can really help jump start a relationship in a lot of situations.
Could you tell me about how you got to where you are?
In other words, you're asking someone to share their story. It's usually insightful, maybe inspiring, but everyone has a story and often people are not afraid to share it. There is also something personal about hearing about the long and sometimes windy trail that has lead to their current endeavors. This question can be very natural especially if you're young in your career or a student.
I interned at Adobe when I was an undergrad. I met with an executive in another department to get some advice as I was still figuring out what I wanted to do for my career. I asked that one question (with a few follow up questions, of course) and we talked for 30 minutes. At the end of the conversation she asked if she could help me by introducing me to other people in the company. She asked me! There is something about telling someone your own story that often inspires you to lend a hand. Now, I'm not writing this so you can ask this question and try to flatter someone into helping you. That is not my intention. I just want you to know that it's a common outcome to asking this question. The most valuable thing you'll get out of asking that question is getting to know someone better and being inspired by their story.