To keep it simple lets stick with the standard national hive.
The hive body boxes come in 2 sizes, 'deep' or 'shallow'. The deep boxes are generally used to hold the brood, and are therefore often called 'brood boxes'. The shallows are generally used to hold stored honey, above the brood box (and with a queen excluder to stop her going up into it and laying eggs). Because these are placed above the hive, they are called 'supers'.
An eke is a much smaller box, so called as it allowed/allows a few extra inches of comb height and therefore allowed the beekeeper to 'eke out' a bit more honey from each comb. Having said that, we dont generally use ekes for that purpose, we tend to use them to allow us to put something on top of a deep or shallow box without it being crushed by the bottom of the box above (with frames in). An example is an apiguard tray, but they can also be used to feed fondant, or pollen pattys, or use other varroa treatments like apilife var strips.
In other words, its a useful bit of kit, and the reason we use an eke where possible, as opposed to an empty shallow box, is that there is less empty space added for the bees to keep warm.
Underneath the brood/deep you obviously have a floor (on a stand maybe). The top of the hive is capped off with the crown board, it sits on the crown of the hive. Technically speaking a crown board is just a plain board (or sheet of perspex!), however some come with holes in which allow us to both feed through those holes by placing a feeder above them (and not disrupting the hive), and they can also be used to clear bees out of a super which we want to extract. By putting 'porter escapes' in the holes, and the super full of frames of honey above the crownboard, the bees on those frames will want to get back inside the hive to the pheramones - and the porter escapes allow them to go into the hive, but not come back through. In theory!
