An option on a depositary receipt gives the holder the right, but not the obligation, to buy (call option) or sell (put option) a depositary receipt at a predetermined price within a certain period.
Example: Suppose you purchased a call option on a depositary receipt of company X for €10, with a strike price of €50 and an expiration in six months. If, after four months, the share certificate has a market price of €70, you can exercise your option right. This means you get to buy the depositary receipts for €50 when it's worth €70 on the market, making an instant profit of €20 (minus the €10 you paid for the option and any other costs). If the market price of the depositary receipt never exceeds €50 before expiration, it will likely not be beneficial to exercise the option, and you lose the initial €10 you paid for it.