What pipe/adapter do I need to make this sink connect to this P trap?
You'll need to install a drain assembly. In a bathroom, pop-up drain assemblies are common.



Here's an image of the drain assembly, with all the parts labeled.


(source: naturalhandyman.com)  

The tailpiece of the drain assembly, should attach to the trap assembly. If the trap is too low, an extension can be used.

When installing the drain, put a bead of plumber's putty around the drain hole of the sink. Then bed the drain flange in the plumber's putty.  From below, install the washer and nut, and thread on the tailpiece. When you tighten it down, the excess plumber's putty will squeeze out. Use you finger, or a rag to remove whatever oozes out.