That thing is called CV Constant velocity Joint.
I recently replaced mine, i got them on rockauto.com each was shipped in an individual box 50$ for each or so, so for 100$ u replace both. Also supposed to send em back the old ones for 20$ cash on return, but i dont know where to send em to... they didnt include the instructions how to return old ones... but the SHIPPING was FAST! They arrived in 3 days. So i higly recommend it.
To remove em u u gotta drop out the old ones, by prying em out of the tranny, and before that release the outer part of the CV joint. On my car u take the wheel off, then the HUB wheel nut and then take the 3 screws on the most bottom of the car out that disconnect the hub with the brakes from the arm that pushes on the round joint below the brake caliper.\
That allows u to pull the brake caliper and whole hub to the side, without disconnecting anything major, then the CV simply drops out. U can do this in reverse and stuff, its preaty simple. This can also be done with the WHEEL still attached, but i find it eazier to see if its gone.
The rebuild or new CV's are much better thne the junkyard ones, The rubber BOOT at where it flexes is filled with Grease and properly clamped with a round crimp that holds the boot on. So its very well made, u just POP it in and thats all! Haynes manual says to replace em every 100000 miles my car has 160000 and the boots were gone.
Anyways hope that helps. get new ones, its worth it.
Word of advice, make sure u DONT BANG on it with a hammer, because it has threading on the wheel hub side. I damaged mine made putting the nut on it very HARD. Also Drive the inside part into the transmission, ALL THE WAY. What i did is simply stick it in, and then BANG on the hub side with a hammer through a peice of wood, so the threading doesnt get damaged again. Also MAKE SURE ur CV joints look the same as the ones in the picture, they have different ones, for some reason at least for a tercel.