If your only objective is to tighten the drain:
- Remove the trap
- Remove the square grate above
- Use a basket wrench to attempt to tighten the drain from above.
- If the tail piece turns with it, try grabbing it with your hand. If your arms aren't long enough you'll need an assistant.
- If you need more strength on the tail, get a strap wrench, or fake one by wrapping a cut up bicycle inner tube around the tail and using channel locks. Either way be gentle ... the tail is soft and fragile.
You may either turn the basket and hold the tail piece, or the other way around. One way may be easier.
The tail piece is screwed into inner threads in the bottom half of the drain so the above ought to be enough. If the top and bottom halves of the drain are seized together you'll have to remove and replace the whole thing. But if it has become loose it's probably not seized together.
If you want to remove the drain
Follow all the steps above but turn the other way. You may however find that the tail piece will unscrew from the drain. It seems to be screwed in to the drain's inner thread. Like most drains, it's probably firmly stuck in there but if you're (un)lucky it will come out, and you'll be left with just the stubby drain sticking out from the wood.
- Try the strap wrench or inner tube on the drain itself. There's not much sticking out and you don't want to damage the threads but it's worth a shot.
- If that doesn't work you can grab the bottom of the drain from inside it using an expansion wrench or a drain key or a cam wrench. Pick one that you think might be useful for at least one more project in your lifetime. :)
- If the parts are really seized up use a mini hacksaw to cut the drain in half from the inside, being careful not to damage the sink. Then replace it. Again, if it has come loose, that's a good sign you won't have this problem.
Or do it the way it was probably installed:
My guess: After you remove the tail piece you can slice through any silicone or glue holding the basin to the counter, remove the entire basin, replace the drain then put it back. That may be the easiest approach especially if the tail piece can be easily removed with a strap wrench.
Replacing the tail piece
I suggest you replace the tail piece with a more typical one that uses a compression nut on the drain's outer threads, rather than one that screws into the inner threads. Because ..... it's a lot easier to remove later and less prone to seizing up.