Can I use a 3-wire AC adapter in a DC circuit?
Read the label on the power supply and it should be no surprise there are three wires.  Note that this supply doesn't put out a single voltage, which would require two wires.  The label clearly shows that this supply puts out 5 V and 12 V, hence it needs at least three wires.  Usually when a supply puts out multiple voltages, they have a common ground, which is also the case here.  That is why there are only three wires instead of four.

Black is almost certainly the common ground.  Use a meter to see which one is 5 V and which one is 12 V.