You know how those porcelain bathroom sinks have a little hole on the back wall? Like here, in this photo, there’s a little slot above the “Shanks” lettering?
What’s that hole for?
It’s to prevent the sink from overflowing. It leads to the drainpipe, bypassing the stopper at the bottom of the sink. This way, if you have the stopper closed, and the water running, and are called away (to answer the phone, say) when the water reaches the hole, additional water will drain away, instead of overflowing out of the sink or tub.
It seems like it should be designed to prevent overflow when the sink plugs up, but it won’t. It’s exit hole is right where a hair plug would start, just below the entrance to the drain hole.
I think it’s mainly so that if you fill the sink with water and then add too much other things (like laundry to wash) that when the level rises it won’t overflow.
But it’s too wimpy to take much overflow at once, such as if you dunk your face into the water it will overflow anyway.
So, all in all, it appears to be useless, just a remnant of victorian design that will never go away because plumbers are not great analytical thinkers. Plumbing hasn’t changed for the better in a century, just different styles.
Like everyone else said it’s a overflow hole. But it’s also a vent. It’ll help water make it down the drain without ‘bubbling’ (you know the gulp gulp sound that happens when you use a water cooler). Of course you’re entire house has (well should have) at least one big drain stack that comes out of your roof that does the exact same thing for the entire house.
I’m going to guess it’s a razor blade disposal slot.
Old bathrooms often had a slot in the wall for men to safely dispose of their used razor blades ( image ) . It looks to me like your sink has one built-in.
Good one DakotaDog.
Er … I think you’re missing a 
there Dakota?
(Ya never know, someone might believe it, and clog up their pipes with used blades …)
It is NOT a razor blade slot! Those things are on the wall, usually inside a medicine cabinet.
I am absolutely aghast that people have no idea what an overflow drain is. There’s on on your tub too.