Closure is a term from mathematics and computer science. We could "dumb it down" and dream up another name (like union or merge) but it wouldn't help.
Closure essentially means a list all objects gathered up from "below". An example will help.
The hierarchy of reports can be rather large (accounts and journals too). There are several types of reports and within each type there could be many different reports. The obvious way to explore the hierarchy is to activate a folder and display/edit a list of objects in the folder. Another way to explore is to activate a folder and see a list of all objects in the folder, plus all objects from all folders "below" (i.e. a list of everything from the active folder "down"). Closure indexes achieve this.
There are two basic types of indexes in NewViews:
Direct indexes include only the items immediately below. The sort by column title will display a single chevron (v for A-Z sort or ^ for Z-A sort).
Closure indexes include the items immediately below, plus all items to the "bottom" of the hierarchy. The sort by column title will display a double chevron (vv for A-Z sort or ^^ for Z-A sort).