said by robbin:
You don't need to turn off any water. Still it would be a good idea to turn off the valves under the sink so the wife doesn't step over you while you are working on it and turns the faucet on.
What he said, but hopefully you can leave those valves alone as turning them can open up a whole 'nother world of plumbing fun.
You don't have to mess with your water supply valves because you're working on the water drain, an entirely, physically separate system.
Do try a sink plunger first, it's likely that's all you need. What I don't see in your picture is the stopper lever (the gizmo that you use to lower the drain's stopper (plug) in the sink when you want to fill the sink with water). Does your sink have one, or do you use a rubber plug to close the drain?
The reason I ask is that sometimes a big glob of slimy hair will build up on the lever's arm that's inside the drain pipe. First thing I usually do if simple plunging doesn't free up the drain is to remove that lever, clean it off, re-try the pointy plastic cleaner and then remove the sink stopper, put the lever back in and plunge it again.
Assuming it's clear then, take the lever back out, drop in the stopper and put the lever BACK in the drain (lined up so it catches the hole in the plug's stem) and you're back in business.
Shit, I made it sound complicated, it's not! Try the plunger first.
OH, and before you plunge, put duct tape over the sink's emergency drain so that the air/water you're trying to force down the drain won't just shoot out of that emergency route and go all over your hands.