Tattoos of tears falling from the corner of the eye are said to mean that the wearer has killed somebody, or that someone close to them died, and can indicate affiliations with Gangs. The teardrop tattoo originated from Chicano gangs in California. These are typically Jailhouse tattoos (and thus have a broad range of meanings, often local to the prison they were done in). These tattoos have become more popular in recent times, straying from the original meaning.
They are also known as a "Cholo Tear".