Died: Season 1, Episode 5, "You're Undead to Me"
Method of death: Neck snapped by Damon (Ian Somerhalder) when he tried to stop his "nephew" from escaping the Salvatore basement.

"That was to come right out of the gate and really shock people with what Damon was capable of and willing to do and in a pinch. This idea that even family meant nothing to him that early in the season as we're dealing with the relationship between Stefan (Paul Wesley) and Damon, the brothers. The idea that he could so coldly and without remorse delete a member of his own family really told the audience a lot about who Damon was."

To cover up Zach's death, Damon told the townspeople that he was on vacation, so they probably "think that Zach is off on an ashram in Asia somewhere meditating."