What type of pipe joint is this, and how do I repair it?
That's known as a slip joint. If you remove the slip nut, you should find a beveled washer on the pipe. 

To put the joint together, you'll slide the slip nut onto the upper pipe, followed by the beveled washer (bevel side down). Then you'll slip the upper pipe inside the lower pipe, and use the nut to tighten the joint. 

Looks like it's a waste tee similar to this.



The trap should be attached to the bottom of this piece, probably using another slip joint.

When tightening this type of joint, you want to be careful not to overtightened. Hand tight is usually all it takes.  It should be just tight enough, so that the pipes don't easily come apart.