I guess it would be overwhelming to a new-comer to horse clothing when sorting through all of these different things! So sheets are lighter weight than blankets and just used to keep the horse warm in cooler temperatures (people vary drastically in what temperatures they use sheets and blankets on based on the climate you live in, age of the horse, personal preference, etc). For example, I use a sheet when it's 40-50 degrees, a blanket when it's 30-40 degrees and put both the sheet and blanket on if it's below 30.
Now, if something is called a stable sheet or blanket, this implies that it's not tough enough to be worn by a horse that is turned out and thus, should only be worn in the stable. Oftentimes, these are also lighter weight than turnout sheets or blankets (since it's warmer and out of the elements in a barn). "Turnouts" on the other hand, imply that they are tough enough to be worn while a horse is turned out and also generally implies that they are thicker/ warmer than stable sheets/ blankets. A general rule of thumb should a sheet or blanket not state whether it's a "turnout" or for the stable---turnouts are usually made of tougher material, are bulkier (particularly blankets, i.e. warmer), are waterproof and have legstraps.
As for fleeces, this is a third layer that your horse could potentially wear in extremely cold weather. For example, I will use a fleece if it gets below 20 degrees. Think of this as the fleece liner you can zip in and out of your own jacket. The fleece is worn under the sheet and blanket and some fleeces don't even have straps. Because of the lack of straps and utter lack of waterproofing, fleeces are generally only used when horses are staying in (say if they come in overnight).
Now for all of the extras: COOLERS are used in the winter on a wet horse to keep them warm. For example, if your horse was particularly sweaty after a ride, or you needed to bathe or sponge him off in cooler weather, you would put a cooler on them immediately after you bathed or untacked them while in their stall (i.e. they are not to be turned out in these). Specifically coolers are made out of wool or fleecey type material and will aid in wicking some moisture away while keeping your horse warm. You generally never keep coolers on for long periods of time, just long enough for your horse to dry/ dry enough to change him back into a sheet or blanket. On a separate note, if you show, you will often see riders with coolers draped over their horse's hindquarters (and perhaps covering the rider's legs too!) and this is just to keep the horse's muscles warm while waiting to go into the ring.
Now, ANTI-SWEAT SHEETS are generally used in warmer weather, but could also be used in cool weather. These sheets are simply designed to wick water away and thus are used after a horse has been hosed off or is sweaty and allows the horse to dry. They are generally considered to be more breathable and lighter-weight than coolers, thus the reason coolers are more ideal for cold weather. Personally, I just have a cooler and don't worry about the antisweat, as I figure my horse will dry quickly in warm weather thanks to the heat and sun!
Now for QUARTER-SHEETS. These are sheets (often fleece or wool) that are designed to fit under or around your saddle which you can ride in and only cover your horse's hind-end and back. Some quarter sheets are designed to go under and attach to your saddle, much like a saddle pad, while others are designed to come up over the rider's legs and be secured over the rider. Personally I like the idea of the ones that fit under the saddle, as it doesn't seem so safe to me to have something over your legs/ attached to the horse! Quarter-sheets are great if you have to ride in especially cold temperatures, as it allows your horse and his muscles to stay warm, allowing for more comfort, but more importantly, also preventing potential muscle strain due to working a horse with cold muscles. The only drawback is that some horses have to get used to the sheet flapping around a bit while being ridden (although I have never experienced this) and most folks say to never jump in a quarter sheet as it could accidentally get caught on a jump/ standard mid-air and/or may flap around more than normal, potentially spooking the horse.