CV joint

It will be easier to fit a new shaft, doing it at home can be trick as you may have trouble undoing the shaft end nut, do you have a get jack and stands as well, if you want to do this, let me know and I will send the method.

It's called EZ Boot or Speedie-Boot. With these all you have to do is cut the old boot and clamps off, clean off any contaminated grease, fit the new boot, refill with grease, tighten the clamps and bolts, and you're done. BUT, if the old boot has been leaking for awhile, you need to clean the joint and check it for pits or grooves. If the joint is in good condition, just replace the boot. If not, replace the axle shaft.
Remove the wheel. Remove the axle nut. Remove the caliper and lower ball joint. Use the correct tool to separate the axle and spindle.

Most shops and mechanics won't recommend this type of repair. It is slightly less reliable than custom-fit boots, but the labor and parts costs are much lower than either replacing the joint/half-shaft or using a custom-fit boot. They can't charge you as much for a EZ Boot/Speedie-Boot.