* *sterben* is neutral and can be used for every kind of death. Also this is by far the most used.

* *ableben* is a little bit elevated or euphemistic. It is usually used for quiet and slow deaths, i.e., by illness or old age. For example, this would usually not be used for somebody killed by an explosion.

* *versterben* – if used in inofficial context: same as *ableben,* except for the euphemistic touch. If used in official context: same as *sterben.*

* *abtreten* actually means *to resign, to surrender one’s post* or *to exit.* It is only used colloquially in the context of dying and at least with a grain of irony or metaphor. 

* *erlöschen* is only used for dying in combination with *Leben.* In this case this a very formal style.
> Sein Leben erlosch. – He died.

* *krepieren* has a disrespective note and also implies a painful or miserable death.
> Krepier, du Schwein! – Die, you bastard!

* *verenden* is usually only used for animals. If used for humans, this implies disrespect for the dying person.

* *eingehen* is almost only used for plants and lower animals. I have never seen it used for humans.