How to fix this leaky sink drain?
That is commonly called a kitchen sink strainer or basket strainer. 

You should start by soaking the parts that you can see in the picture with a penetrating spray lube, the kind that says "helps loosen rusted parts".

You will need to disconnect the plumbing from the strainer by removing both the metal and the plastic slip-joint nuts. Then remove the strainer by unscrewing the large metal ring nut with the little "nubs" all around the outside. This will likely be difficult. I usually use a large long-handled screwdriver and a hammer to try to back the nut off. If this does not work, you may have to actually destroy the ring nut to remove it (not uncommon).

Take all the parts down to the local mom and pop plumbing or hardware store and they will hook you up (don't even bother with a "big box" hardware store).
You will need:


new basket strainer
1 lb. plumber's putty
at least (1) 1 1/2" SJ washer


The rest of what you need should come with the strainer.

NOTE: this leaked because the plumbers putty seal between the sink and the strainer's upper flange failed. Caulking plays NO part in this assembly.