My rear differential is broken, a piece of one part broke off. My mechanic tells me that I need to replace the entire differential. Is that true or can I simply replace the broken piece?
Once a piece breaks off it can grind against the others, damaging them. While diffs can be repaired, it is much easier to just replace the whole thing with one from a junk yard, shouldn’t be hard to find for your jeep. I did it in a parking lot to my '72 Duster in a day.
Most likely, your mechanic is correct. A broken-off gear tooth, for example, can do an awful lot of damage to everything else in the diff. There is a lot of labor involved in properly rebuilding a differential, and the price of individual parts can be very high. The most cost-effective fix is often a complete axle assembly (including the diff) from a salvage yard. However, your post is a wee bit short on information, like what kind of car you have.
1997 Jeep Wrangler
Oops! Just saw the tag on your original post.
Yep, a salvage yard axle assembly is the way to go, IMHO. But get an estimate for this as well as for rebuilding your current diff.