First off there's the hierarchy. It's worth knowing a bit. It goes like this
Order
Family
Genus
Species
There are some levels above order and one or two extras like subclass, but these are the most important ones for now.
The Mycokey wheels and Fungi of Temperate Europe, which the wheels are from, have a huge section under Agarics, 700 pages. This is the first wheel, and you can see gilled = Agarics in the bottom left, ie second item.

Wiki says: "An agaric is a type of mushroom fungus fruiting body characterized by the presence of a pileus (cap) that is clearly differentiated from the stipe (stalk), with lamellae (gills) on the underside of the pileus. "Agaric" can also refer to a basidiomycete species characterized by an agaric-type fruiting body." .... "Most species of agarics are within order Agaricales..."
So, roughly speaking, if a mushroom has gills it is in the order Agaricales. This is divided into families. These are easy to spot because they all end in "aceae". Eg:
Agaricaceae
Amanitaceae
Marasmiaceae
The family is split into various genera (plural of genus). Agaricaceae family includes
Agaricus (the type genus the family is named after)
Barcheria
Bovista
and other genera
Wild Food UK lists all mushrooms in alpabetical order of the latin name. The Latin name is always two words. The first is the genus and the second is the species. So looking at this list you immediately see the first 11 are Agaricus. Obviously these will have similar attributes. 9 are edible and two are poisonous, so you need to know how to tell them apart.
Here are the main Agarics wheels



