A potential verb is a compound conjugation of a base verb with a helper verb expressing the potential (possibility of doing verb action exists) of that base verb. It can usually be translated with English phrases such as "can" or "be able to".
CJ potential add -る to 未然形
long potential add -れる to 未然形
short potential add -得る to 連用形 often the -いえる contracts to -える (eg いきえる > いける)
(common forms that never contract: 有り得る)
CJ potential add -らる to 未然形
long potential add -られる to 未然形
short potenital add れる to 連用形
Earlier texts use -ら and -らゆ as a potential form instead of -る and -らる. Which of these forms came first is not known nor exactly how they were formed.
However, it is almost certain that forming the potential utilizes and 得る(う) (to get, obtain) and for some forms possibly 有る (to be).
The long potential is the same as the CJ potential with the following changes having taken place.
Modern Japanese abandoned the use of the 終止形 verb form and uses the 連体形 as a predicative (for "ending" sentences) instead. Forming the 連体形 adds a -る to the 終止形. Moreover, modern Japanese reduced the conjugation classes verbs belong to, resulting in a change of -得(う)る to -得(え)る.
The origin of the short potential is quite clear though. Godan verbs simply add 得る (to get, obtain) to the 連用形. For most verbs, the イ sound of the renyoukei contracts with the エ sound of 得る as descibed in formation.
The short potential of Ichidan verbs is likely an analogy to Godan verbs.
The CJ potential is not used normally anymore except for literary purposes &c. For Ichidan verbs, the short potential is considered colloquial and not standard usage. For Godan verbs, both forms are possible. The long potential is considered more formal however.
It may be that the using the short potential is influenced by the fact that the long potential is ambigous; it may also express a passive or a honorific; the short potential is unambigious. Preference might be given to the short potential to avoid this ambiguity.