What happens to the root from an extracted tooth that is broken off and is still imbedded in the gums?

Depends. If it is a very small fragment that was left because high risk for other complication by going after it (nerve damage or sinus perf) then it is not uncommon to have to leave it and it may abcess but more likley it will eventually surface or stay where it is in the bone. If it is large it needs to be removed.
It depends. Sometimes a broken root in the gums will "work it's way out.' if the root is still embedded in the bone, it may remain a while. In either case, this root can also cause an infection. We recommend a root tip be removed so you do not risk further problems! In addition, it can further decay, making it's later removal more difficult.
ROOT BROKEN. The broken root should be taken out or it may cause infection or abscess (foreign body) in the gum.