When someone avoids eye contact, it may mean they don't want something about them to be seen. They could be lying, masking their emotions or insecure in some way. Avoiding eye contact is typically related to a fear of rejection.
Eye contact is vital in communication with the world in general and in intimacy with loved ones. It shows attentiveness and the desire to be known. Personalities that are more confident and outgoing have less of a tendency to avoid eye contact. People who make regular eye contact are also seen as more trustworthy and affectionate. Low self-esteem and introversion may contribute to poor eye contact.