A double plural is a word whose root is twice modified by a plural; in English, this usually means two suffixes or an irregular plural form plus a suffix. In moderns times, the suffix in question is almost always a plural -s added onto an existing plural that does not use a plural -s.
This can often be a grammatically correct choice, especially when using a mass noun. For example, fishes is the correct way to refer to multiple different types of fish; sands is an acceptable way to refer to either multiple types of sand or poetic concepts (shifting sands, sands of time).
Just as often, a double plural comes about because the language is changing to the point where the original plural is no longer recognized by many speakers. For example, the Old English word cild was originally used for both singular and the plural, but was then pluralized into cildru, and later repluralized into children. More modern examples include the insigne being absorbed into its own plural, insignia, and then the emergence of a new plural form, insignias; candelabrum being assimilated into candelabra and spawning candelabras; and medium being overtaken by media, resulting in ‘medias’.
Some double plurals which are currently struggling to emerge into the English language are ‘dices’, ‘bacterias’, ‘cactuses’, ‘shrimps’, and ‘quails’. There are also ongoing attempts to clarify the plural you, with common contenders being y’all and youse.