Sunday morning plumbing problem?

There should be a plastic gasket that encircles the tailpiece(drainpipe that attaches to the sink drain), and a nut above it that is to thread on the trap threads and creates a waterproof seal. It should be that the gasket is tapered on the bottom side to allow it to contact the trap pipe at its opening. Notice the way it is to fit that way? Loosen the other nut on the opposite side of the trap if you need to adjust the height so water will drain properly.

Looky here, the paper gasket is called a friction ring. It goes between the rubber gasket and the big round locknut that secures the strainer body to the sink. Beneath that is the smaller threaded section that the tailpiece (aka drainpipe) attaches to using a top hat washer (flat brim with ring that sets inside the tail piece) and a coupling nut. The tail piece should have a wide brim on the top for the washer to seal against and for the coupling nut to push up against to create the seal. When making up the drain that nut should've gone on first then the lower coupling nut for the P trap.

The top locknut and coupling nut might of cracked from being over tightened and the hot/cold cycles finished those off. You can buy those individually at the big boxes and hardware stores. The trick for tightening stuff like that is to get it snug then run water thru the drains. If nothing leaks then leave it alone. If drips occur then just tighten enough to stop those.

Try self amalgamating adhesive tape as a temporary fix. This stuff is waterproof and bonds to itself very effectively. Available in all major DIY stores. When you apply it you stretch it to about twice it's length. It then bonds tightly. Any other tape will fall off due to the water. See link for an example.