What kind of gate(s) can I use for this logic?
For a circuit like this with only two inputs, it would make this a little easier to analyze if you expand the X:

IN-1 IN-2 OUT
0    0    0
0    1    1
1    0    0
1    1    0


Since there is only one term resulting in a "1", you can write a Boolean expression describing solely that line. In this case, this means (!IN-1) AND IN-2. You can construct a circuit to compute this using an AND gate, with an inverter on the IN-1 input.