(FG) Firm ground boots
Firm ground soles are designed for use on natural grass pitches where the grass is kept short. Ideally for use when the pitch is dry, they can handle slightly damp pitches too though.
Most modern firm ground football boots are equally at home on 3G & 4G pitches.
Boots with a firm ground sole usually have plastic or hard rubber "molded" studs. Unlike standard boots with six studs, firm ground soleplates tend to have multiple studs of various shapes & sizes.
Firm ground boots are not recommended to be used on traditional sand-based turf pitches. Although they work well enough on them, they're not really designed to. Using them on old-school turf can wear the studs down faster than usual.
(SG) Soft ground boots
Soft ground soles are designed for damp, muddy pitches with long or short grass. They're used when you need to get the most amount of traction possible and are worn by most players during the winter months.
Soft ground boots have the most traditional form of sole, featuring six conical studs that screw into the soleplate. That said, modern styles are also prone to going for a "mixed" sole that also includes a few molded studs as well.
The traditional studs are usually metal, though other materials can be used. Most of the time they are replaceable on soft ground boots, meaning they could potentially last you forever.
Soft ground boots are not recommended to be used on a bouncy firm pitch or 3G pitches.
(AG) Artificial grass boots
Artificial grass boots are a modern idea that came about because of the rise in popularity of 3G turf pitches.
AG boots are meant to look & feel like a firm ground boot, however they offer increased grip and cushioning that's needed on 3G.
At first glance the studs will look just like the ones you'd find on a firm ground soleplate, however AG boots tend to have the studs hollowed out. This not only keeps them lightweight, but it's how they absorb the harsh impact that comes from running on 3G.
Artificial ground boots are not recommended to be used on grass pitches.
(TF) Turf trainers
Football trainers that are designed for the small-sided version of the game. They are meant to be used on water or sand-based artificial grass pitches. They can also be used on 3G pitches too, though they don't offer as much grip as AG boots.
They're very easy to recognise thanks to their soleplate consisting of multiple rubber pellet-like studs.
Turf trainers are not recommended to be used on indoor pitches. Lots of people mistake turf trainers for indoor shoes because they do kind of work indoors.
(IN) Indoor shoes
These are used when playing indoor football.
They can look like the usual Nike trainers & they can also look like regular football boots with a flat rubber sole. The main difference between indoor shoes and turf trainers is that the rubber sole is flat (with maybe a couple of slightly raised rubber portions).
Turf trainers are not recommended to be used on anything except indoor courts. You'll slide everywhere on grass and you'll ruin them on 3G or turf.