A few things here. First, you need to remove ALL the mold. Second, you need to prevent it from coming back. Third, I would fix that grout in the tile on the corner to prevent splashes from getting in and causing mold in there too.
You're going to need to remove the faucet completely. The hole that that thing is set in is most likely a mold farm. Just shut off the water, unhook the hoses and undo the nut clamping it to the bottom. The tap should come out the top no problem. Wear a mask, I'm expecting the worst for that one.
Now you're going to have to clean that mold, all of it. If it's caked on, chisel it off with a screwdriver or something, then get some sandpaper and make sure it's all gone. Check under the countertop too, there might be some down there. If it was me, I'd take the whole sink out and inspect, but that's probably overkill. However, if the caulking around the sink isn't very good, do go head and disconnect the drain pipes (there's a flange that unscrews) and pop the sink out.
Now, we're all clean, good. We have to fix the color. I'd get some very fine sandpaper and disregard the finish of the wood, it's already going to need repairs. Clean the mold off with the sandpaper, then get a fresh piece of sandpaper, sand down a bit more, then a really really fine grit sandpaper and do a final sanding. With the really really fine stuff, you might be able to feather the edge of the stain. If not, that's fine. You should have a good area of plain wood to work with, and the faucet should still be out.
Now, I'm not sure if that countertop is stained or if it's oiled. It's most likely oiled, and if that's the case, go ahead and do some research on how to replace the layer of oil. If you want, perhaps explore doing a layer of urethane, that will prevent mold very well, but I don't know about the aesthetics.
Anyways, now we have no mold and a fresh canvas to work on. If you took the sink out, make sure to caulk the edges very well when putting it back in. When you put the faucet back in, put a big fat bead around the hole before you replace it. Quickly tighten the nut a bit below, then get above and clean the edge of the caulk off with your finger. This will seal it so you won't have any more mold getting in the crack.
At that point, just make sure to keep the area dry to prevent future mold, or have a preventative coating on the wood. You could always bring the tile out a bit too, just for that square area of the countertop. If you decide to do that, make sure to install the tiles properly and to do it before you do put the faucet back in.