I met someone for the first time the other day and when I shook her hand she didn't look me in the eyes. It's amazing what a difference such a seemingly minor non-verbal signal makes.
If you don't look someone in the eyes when you meet them, there are a number of assumptions the person could make:
1. You're hiding something
2. You're very insecure
3. You feel intimidated
4. You're not really paying attention
5. You're just rude
None of these assumptions are positive, and hopefully none of them are true. In my situation, I could tell by the rest of the person's body language that she was just a bit insecure. But I could easily see how other people could read her signals as being rude.
The worst part is that I don't even think she realized the message she was sending. It's very possible that she has had a hard time connecting with others and not known why. Think of how many relationships have started off on the wrong foot because of this small mistake!
When you meet someone for the first time, eye contact is just as important, if not more important than a firm handshake. Make sure you make the best possible first impression and look your new acquaintance directly in the eyes.