The other answers discussed damage to a chip through ESD. Latchup is a slightly different phenomenon. Latchup usually happens when you take an input (or output) of an IC, and put a voltage either higher than VCC or lower than Ground on it.
When this happens, because of the way IC's are made, there are all kinds of unintentional paths that current can flow in (and basically, a whole bunch of transistors and diodes appear in the circuit that don't have any effect when the circuit is operating normally. This can cause an output to get "stuck" in one particular state, or for an input to go dead.
Latchup isn't always fatal to a chip; sometimes it will fix itself if you turn the power off and then back on again. (In other cases, it can do permanent damage and destroy the chip).
ESD (even if it's far away from the circuit itself, such as zapping the case of an instrument) can cause circuit voltages to spike (either positive or negative) driving the signals beyond their legal ranges and triggering latchup.