An eyelash curler is a hand-operated mechanical device which has a set of grips that operate a clamp. The grips are held and used similar to scissor grips. They are, unsurprisingly, used to curl eyelashes. They're typically made of steel.
An eyelash curler should not be confused with a mascara applicator (sometimes called a wand) which has short bristles on a small stick and typically has a colorful plastic handle. As a general rule-of-thumb, if it doesn't look like a medieval torture device, it's probably not an eyelash curler.