I have a belay device that looks like:
which I used when I am belaying someone who is top roping or leading. However, my rope only goes through one of those slots. What is the purpose of the other one?
Most belay devices come with two holes for rope, for various reasons:
Some devices are asymmetric and the two holes provide different friction
Others are shaped with an up and down orientation, so two holes are provided to make it easy for left/right handed people
Some lead climbs are better tackled with two lines of anchors. The two holes make it easy to belay a two-rope climb. This usually needs a lot of concentration from the belayer and strong communication between belayer and climber
As Ryley mentions - it can be used as an arrestor for rapelling/abseiling by looping the rope around an anchor and putting a loop from each length through each eye
Your specific device (it is a symmetric design) would be best suited for scenarios 3 and 4.
It's a trick! Don't use it!
Rappelling/Abseiling is the answer though.
Some people use it for belaying twin or double ropes too.