My favourite definition of "awkwardness" comes from Steven Pinker: it's the uncomfortable feeling you get when you realize that your concept of your relationship with someone else doesn't match their concept. The intensity of awkwardness roughly corresponds to the magnitude of difference in relationship concepts.

For example, I was at a dinner party recently and greeted a female friend with a handshake. She said, "That was formal," and slight awkwardness ensued. When we were leaving I made sure to give her a hug.

If, at the end of dinner, someone had pulled out their wallet and starting putting money on the table to pay for their meal, the awkwardness would have been much more intense.