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 -ru to mizenkei
long potential add -reru to mizenkei
short potential add -eru to renyoukei often the -ieru contracts to -eru
(common forms that never contract: arieru)
CJ potential add -raru to mizenkei
long potential add -rareru to mizenkei
short potenital change last syllable to e-row and add -ru
Earlier texts use -ra and -rayu as a potential form instead of -ru and -raru. 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 uru (to get, obtain) and for some forms possibly aru (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 shuushikei verb form and uses the rentaikei as a predicative (for "ending" sentences) instead. Forming the rentaikei adds a -ru to the shuushikei. Moreover, modern Japanese reduced the conjugation classes verbs belong to, resulting in a change of -uru to -eru.
The origin of the short potential is quite clear though. Godan verbs simply add eru (to get, obtain) to the renyoukei. For most verbs, the i sound of the renyoukei contracts with the e sound of eru 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.