There are two definitions of enumeration, or enumerating details: to list details and to count details. This example covers the definitions in itself, in that it's both listing the two definitions while also counting them. Enumeration as a term is used most commonly in mathematics and not in laymen usage, although nearly all aspects of everyday life display signs of enumeration in the form of lists, such as ingredients.
Enumerating a set of details may create a list, but it does not imply that the list needs to be in sequential order. The definition of enumeration is very wide and subject to the needs of those using it at the time. As a result of this, lists can be given any kind of order, whether patterned or arbitrary, and however abstract the creator wishes to make them.
When enumeration is used in mathematics, it is used to list out a set of integers in an established order. This order is simply counting, an assessment of all of the values present within a part of an equation, or establishing a list so that the values are known in a sensible order for other uses.
When used for details on an object, enumeration is used to list any aspects of that objects, from how many sides it has to how many colors it has.