You first need to locate the wall studs behind the wall. Then when you look inside the mirror vanity you will see a thicker board along the top edge and perhaps along the bottom edge as well. This is the location in which you should be fastening this vanity. Simply figure out exactly where the wall joists line up when the mirror is set into place, Drill some small holes in the mirror hanging area. And with help using screws that are at least two inches or bit longer, depending upon wall thickness (most are 1/2") put a washer under the screw head and fasten it into the wall.
Ideally you want somewhere around 1-1/2 inch worth of screw into the stud. You can put some bar soap onto the threads of the screws by rubbing the screw on the soap. This will make it easier to fasten the screw into the wood.
A few other tricks. Of course you need help to install this. Put the first screw as close to the middle area where stud is located and do not tighten real hard yet. The mirror will hang on this. Then simply using a level, level the mirror and then do a 2nd screw which will then keep the mirror level. Two or three screws is all that is really needed, but you should put at least two at top and two on bottom area.