I'll offer an alternative to jwh20's answer...
You need to check if your drain looks like that:
This type of drain goes through the entire sink. It is inserted from the top, and the upper part is the "drain" you see when looking into the sink. The rest of the shaft pokes out under the sink, and the black plastic screw goes on that.
If that's the type of drain you got, then the photo indicates the shaft would be broken, which is unlikely. That's a nice type of drain.
Another type of drain is this one:
The top portion goes in the sink from the top, and the bottom goes... on the bottom. The porcelain sink is sandwiched between the two, and they are held together by the screw in the center.
Now this screw often causes trouble because it tends to be flimsy, and sometimes it's threaded into plastic, and if it's not stainless it'll corrode. That's the usual construction on cheap drains.
So I'd recommend first to pull the drain out from the top of the sink. and look at what you got, check if there are broken pieces, etc.
The huge amount of silicone caulk in the pic makes me doubt how well it was installed...
If the drain you pull out has a broken screw dangling from it (or a hole for a screw but no screw) then you'll have to change the whole thing.
First loosen 1, then rotate the whole drain 90°, loosen 2 and pull it out, then you can loosen 3 and recover the metal part of the drain.
Note that normally you clean the drain by putting a bucket under it and loosening #4 and that's it.
Edit
OK looks like t's broken.
You should find a replacement in any bix box home improvement store.
Make sure it has a hole matching the one in the pic, that's where the sink overflow goes.
You should measure the diameter of the threaded shaft (get the same) and the thickness of the sink:
...and/or the length of the shaft, so you don't get one that's too short. "Same length as the previous one before it broke" would be a safe bet.
You can also get the cheapie ones that hold with a screw, but then you still need to measure the thickness of the sink to get one with a long enough screw!
Also make sure you know what controls the plug, it doesn't look like you have one of these little levers on the faucet, that pushes the plug with a system of rods, like that:
...so don't buy one of these, otherwise you'll have nowhere to connect all this stuff.
--
** Quick'n' dirty repair **
Insert blunt kitchen knife as shown by arrow, wiggle it, and pop the drain out of the sink. Don't use anything with sharp edges, like a screwdriver, cause the sink is porcelain.
When it pops out, and you have the two pieces of the broken drain in hand, you'll know what it looks like better than I can descrige so you can get a new one.
Then take all the pieces to a big box improvement store: top part of drain, bottom part of drain, and the trap because you might need to know the diameter of the pipe and it's better to have it on hand.
And then just ask the people there to pick the replacement, and make sure it fits together...
It'll look like this. Pick plastic or metal according to cost.
You may also need to change the trap if the diameter of the new drain doesn't fit the old one. It's not expensive.
Then you need to clean all the junk, especially the silicone goop (just scrape it off with any tool that'll do the job) and screw the new one in place.
