That's a CV boot. It's possible, in theory, to take the CV joint apart and clean it, pack it full of grease, and put a new boot on. What has happened to you is that the boot blew out and the grease has been thrown out due to the rotation. When the grease is gone, water and dirt get in and this is bad for the needle bearings inside. If you take the bearings out and clean them very well, they might not be destroyed to the point that you cannot put the joint back together and get thousands of miles of continued service. It's entirely possible that the bearings are toast, but you cannot tell until you take the joint apart. They make a grease especially for lubricating CV joints, I think it's molybdenum, and you can get it at the auto parts store.

The typical failure mode of a CV joint is that it makes a clicking noise when the steering wheel is turned full lock. Go to a parking lot and turn the wheel all of the way and go in a circle, then go the other way. I'm not certain, but the inside wheel will click if it's the one giving trouble.

Just because the boot is torn does not, in itself, mean that the joint is toast, although it doesn't look good for the joint. If the boot tore a week or a month ago, and you have not driven in the rain, then you should be okay. If the boot tore several months ago and you took the car swimming across the local Montezuma Crossing (a roadway that crosses a creek with water in it), then you might have trouble with cleaning the CV joint. You can take the axle shaft out and then make the decision to put a new one in or rebuild it. If the store will take the replacement back, then you might want to buy it to have on hand in case you need it.