What does a empty teardrop tattoo mean under your left eye?

I don't think the side has much signifigance due to the fact it varies depending on region, nationality, etc.

Here's some meanings for the ones in North America.

Filled in tear drop can signify the loss of a loved one being a family member or close friend.

One that is half filled in, empty at the top means that he or she avenged the murder of a loved one or close friend.

And a empty one can either mean that a loved one or close friend was murdered.

Or that he or she killed somebody.