All of these fittings are quite common and easy to repair. The large gray fitting at the top is the nut that secures the basket strainer to the sink. There is usually a fiber an a rubber gasket between the nut and the bottom of the sink drain hole that seals between the bottom side of the sink and the strainer. On the top side of the sink is the strainer itself that is usually chrome colored (sometimes white or ivory colored to match the sink. The seal between the basket strainer and the top side of the sink is either made with plumber's putty or silicone. Seals made with putty often leak after the putty dries out and begins to deteriorate. It is fairly simple to repair or replace and the parts should be available at any hardware or DIY store.

To replace the strainer you will have to disassemble all of the fittings below the strainer. Then remove the large basket strainer retaining nut. If the strainer is sealed with putty, it should be relatively easy to remove. If it was sealed with silicone you might have to use a rubber mallet or a block of wood and a hammer to dislodge it. Once you remove the strainer basket, clean the drain hole on both sides of the sink and remove any particulate matter. The opening must be clean and smooth to make a good seal.

One you have removed the strainer and cleaned the sink you are ready to install the new basket strainer. The new fitting should come with the strainer, retaining nut and usually 2 gaskets. The gaskets go under the sink between the nut and the sink bottom. I prefer to use clear silicone to make the topside seal between the sink and the new basket. Use a good bead of silicone all the way around the strainer. Place the strainer in the drain hole, then place the rubber gasket on the bottom side first, then the fiber gasket. Next, screw on the retaining nut. Once it is good and tight, quickly clean up the excess silicone on the top side with denatured alcohol. Let the drain sit over night so the silicone sealer can cure. The next day you can reassemble the rest of the drain fittings and use the sink again.

Quickest fix: stop using the sink!

I'm not familiar with this style of waste fitting, but it looks like the grey section is fastened through the sink by half a dozen screws or bolts?

That section is also corroded.

I'd try undoing the screws and looking for a perished rubber or silicon washer that will need replacing.

Sugru!

when you get the new strainer make sure you get a stopper that fits, your old one may or may not fit.(found that one out the hard way).

also when the white plastic pieces drain pieces break (and they will because they get bumped around) you can take the whole thing off and take it to the hardware store to "make" another one with new pieces.

I can't really be for sure from the reflections in the picture but it looks like the water is coming out from under the brass metal nut.

If that is the case, you could try to tighten up the nut to see if it will seal up. Otherwise, the tailpiece white pipe was installed without the nylon tailpiece washer which is like a ring with a lip(L-shape cross-profile instead of the slip-joint tapered profile or flat washers on the other pieces). Which explains the long standing wet conditions to get the black mold going. Normal pipe expansion could have shifted the pipe a bit so just try loosening the nut and recentering the pipe and tighten up.

If the leak is coming above the grey ring, you need a plumber's spanner or wrench with the wide grip to grab those nubs on the ring which is a big nut to tighten down the sink drain and compress the composite felt washers there. You will need plumber's putty and new gaskets to replace if you take apart the sink drainpiece. Of course, that is a difficult job with the sink mounted in place.

You could do with that "as-seen'on-TV" silicone wrap tape to go around the fixture but that is not a good permanent fix.

Good luck.