What tents are used by the military?

I’m not sure why you haven’t received any good answers to this yet. Infantrymen (and those who support them) generally sleep out in their poncho-hooches (a small tarp strung from a tree), but the 70% of the military who work behind the lines (and the infantry when they aren’t actually on the front) sleep in the wide variety of tents in the military inventory.
Poncho “hooch,” from the Japanese word for house. The Brits would call this a “basha,” from an Indian native thatched hut. 20 ounces of waterproof nylon plus some parachute cord or (pro tip) bungee cords. It sets up in seconds, stuffs in a pocket, and keeps the rain off. Notice that it doesn’t touch ground on all sides like a conventional tent; you want to be able to see out on at least two sides, and to be able to get out fast. Notice that one line comes off the hood to make it stick up, protecting the soldier from condensation and leaks. Older ponchos weigh three times as much, and are made from rubberized canvas, but Custom Steel Aluminum Tents Suppliersare lightproof and much warmer, and are much-prized if you can find them.
The Marine Corps experimented with some 2-man tents like this, but the number I have seen for sale at surplus sites makes me think they were phased out quickly. This type of tent is obviously warmer, drier, and has a light-blocking layer so you can use a flashlight inside it, but is a lot to carry. Something like this would definitely be used in arctic conditions, however.

The “Hex-tent,” officially the “5-man Arctic tent. One pole, some stakes, and no floor. If you set the center pole on an MRE box, it raises the roof enough so that you can sit inside on folding chairs around China Custom Pagoda Tents Manufacturersa folding table. Heated with a tiny, wood or gs-burning Yukon stove.
GP small 8-man canvas tent. The standard since WWII, I think. Later tents were manufactured in vinyl, which is waterproof, but very hard to patch, and leaks far more than the canvas model. It’s heated with a “pot-bellied” stove which is prone to explosions, entirely due to overly cautious safety rules (it’s designed to burn gasoline, but fear of fire makes many risk-averse units mandate running it on diesel, which smokes and burns poorly, frequently extinguishing the flame and leaving a pool of super-heated fuel in the bottom of a very hot stove: small boom follows). These should be mostly replaced by now, but I’m sure there are still some around.
We called this the “ARFAB,” although its actual name is “Frame Tent, 16′x16′.” These can be connected together.
A slightly larger, hot weather version of the frame tent, the “Temper Tent” has a raised second tarp above the roof (to diffuse the heat of desert sunlight), although after about a month, the top tarp sags to lie on the roof. It has tube-vents for air conditioning units, which may seem like a luxury. In the desert, one of these tents is significantly hotter than sitting under a tarp outside, and computers and radio will fry if you don’t keep the temperature controlled. I took a digital thermometer into a non-AC tent in Kandahar, and it pegged at 129 Farenheit before failing.
Last and very much least, the “shelter half.” Developed from a Civil War design, the two-piece canvas “shelter half” pup tent may still be used in Basic Training, and soldiers may still be issued them, but nobody uses them, There is no room for equipment, and the tents are too small for the modern average American soldier (WWII troops averaged 5′8″ and 130 pounds, very lean). I have no doubt that the one sergeant major who stubbornly still uses his (and lovingly maintains his black polished boots in his closet, hoping for the day that the Army goes back to them) will downvote me on this, but, no, nobody uses them.

The Model 1864 shelter tent. The soldiers hated it so much that the Army decided to keep using it for another hundred years.

There are also modular command post tents that “boot” into communications vehicles, and other enormous structures like maintenance tents.