There are few different categories used by ATC.
There's approach classification A through D ( or is it E?), based on Vs, pertinent to app and circling minima.
There's the well known wake turbulence categories L M H, and J.
There's the Code A through E, based on wingspan, pertinent to ground ops/parking/twy restrictions.
At our unit, there's also a perfomance-based table of aircraft groups that fit a similar approach speed/missed approach speed (4, altogether), used for determining the distance required for a departing ahead of an arrival in Below Circling conditions. (Also refer to aircraft approach classification.)
There's a similar but separate list used for determining departure separations following a preceding departure. These two lists are similar, but different, and must be consulted as appropriate.
A fair percentage of aircraft are in a common or expected grouping across several different types of classification. (Eg; most 'heavies' are also class D and code E, and would be in the same performance group on our separation lists.) Some are not. Enough that it matters.
I'm guessing other units might well have their own specific purpose-built lists of aircraft groupings.
Please give us the type, if it's requested.
We'll work out which of the various classifications we need apply, depending on the conditions, depending on specific aerodrome restrictions. and depending on other preceding or following traffic.
Thank You.