Chris Manvell says:

The numbers refer to the number of wheels of a steam loco. Thus 4-6-2 indicates that the loco has 4 unpowered wheels, usually free to move from side to side on a bogie, supporting the front of the loco. The 6 indicates the number of driving wheels. Finally, the 2 refers two wheels supporting the cab and, again usually free to swing from side to side. A full description can be found on Wikipedia at https://en.wikipedia.org/wiki/Whyte_notation.
The Whyte system tells you very little about the locomotive. There is a more comprehensive sustem used by many railway systems. The UIC system gives a fuller description of the loco. So our 4-6-2 could be described as 2’C1′ h3 2’2’T35. Numbers indicate the number of un-powered axles, the ‘ indicating that they are free to swivel. The C is 3 powered axles. h4 tells us that the steam is superheated and powers 3 cylinders. Our example has a tender with two 2-axle bogies and a water capacity of 35 cubic metres.
The UIC system is nearly always used to define diesel and electric locos.
Hope that helps.

John Byers says:

A number of excellent answers. To go a little deeper I will add; aside from the wheel arrangement talked about, some steam locomotives have much larger wheels than others. Lower speed locomotives (mostly freight) have relatively small drivers, while high speed locomotives (passenger or high speed freight) have larger drivers. Usually the higher speed locomotives also have 4 pilot wheels in front of the drivers, rather than 0 or 2. The pilot wheels help guide the locomotive into curves and 4 work better than 2 at high speeds. The last set of wheels support the weight of the firebox on modern steam locomotives. Early designs, like the 4-4-0 and 4-6-0 had small fireboxes that fit between the drivers, but as freight cars got heavier and trains longer, larger boilers with wider fireboxes behind the drivers supported by 2 and later 4 wheels became necessary. Switchers usually had no pilot or trailing wheels because they went very slowly and didn’t need pilot wheels, and kept stopping while the crew coupled or uncoupled cars so they didn’t need the larger firebox to keep steam up.
Let’s look at the examples you gave. The 2-8-2 Mikado would have small drivers and would be a freight engine. The 2-8-4 Berkshire (most famous example was the Nickle Plate Berkshires) was an improvement on the Mikado with a larger firebox to heat a larger boiler to be able to pull a heavier train or climb steeper grades. The 2-6-0 Mogul was an old design but used for many years on branch lines pulling very short trains. They were usually small and relatively light weight.
In the 19th Century, in the US, the 4-4-0 American was the standard locomotive for decades. As Passenger train cars transitioned from wood to steel, the 4-4-0 was replaced by the 4-4-2 Atlantic with it’s larger boiler and wider firebox. The 4-6-2 Pacific soon replaced the Atlantic for long passenger trains but the Atlantic was still widely used on commuter trains. A few railroads even went to 4-6-4 Hudsons. 4-8-2’s were less common than 4-8-4 Northerns for very heavy high speed trains. The articulateds (2-8-8-2, 4-8-8-4, etc) were usually used for very heavy freight trains, although there were some very small articulateds used in situations where curves were relatively sharp (such as logging railroads) and a comparable rigid frame locomotive might have had too long a wheelbase for safe operation. 100+ years ago the Erie & Virginian tried some 2-8-8-8-2(& 4) locomotives, called Triplexes for hauling heavy coal trains. The 3rd set of drivers were under the tender. The boiler had trouble making enough steam to power that many wheels and the 3rd set of drivers were removed.
So, if you want a mainline steam freight locomotive as would have been used in the transition era (40s & 50s), you wouldn’t want a Mogul (2-6-0) or Cosolidation (2-8-0) although these might be in use on a branch line, or if you’re modeling 1900-1920. And you wouldn’t use a Pacific (4-6-2) in freight service. If you are modeling a particular railroad you could look at the roster and see when particular locomotives were introduced. Photographs would also help you. If you are free lancing, still looking at rosters and photographs would give you an idea what locomotives were used when & where. And of course some freight locomotives have been used to pull passengers at tourist railroads and on fan trips (in the 60’s the CB&Q ran a Mikado #4960 as well as a Northern #5632 on fan trips. There are more modern examples, but I saw & filmed those in my youth, as they ran relatively near where I lived). And it’s your railroad, you can do whatever strikes your fancy, but most modelers strive for realism and wouldn’t pull mainline freight with an Atlantic, or mainline passenger with a Mikado. Branch lines sometimes used whatever hand me down they could get. Although a large wheeled passenger locomotive with a smaller firebox, would have difficulty pulling a very heavy train, it could pull a short freight on a branch line.

Phillip Collins says:

Most Americans seem to ignore/forget about the Garratt type locos, which were very popular in Africa (e.g S.A., Angola, Rhodesia) but were also found in Spain, Russia (not sure how many), Australia, UK (3 examples, I think) and on many narrow gauge systems (e.g. Welsh Highland, Tasmania). The most popular South African type was the 4-8-2+2-8-4 (4 pilot wheels, 8 coupled drivers, 2 trailing wheels; 2 trailing wheels, 8 coupled drivers, 4 pilot wheels), the two “engines” being joined by the boiler, firebox, cab section. This allowed extreme flexibility, a light axle loading, to a large extent obviated the need to turn locos and made for a very powerful loco with a deep firebox and short, large diameter boiler.