A "sniper" is someone who strikes at a target from an undisclosed location. A sniper rifle, therefore, would be any rifle a sniper uses. But commonly, a "sniper rifle" is the name given to any rifle with inherent accuracy, and modifications made to wring the most accuracy out of it. Most common sniper rifles are built from a Remington 700 bolt action rifle. That gun is readily available at most gun shops as an well made and accurate hunting rifle.
Modifications may include a cryo-stress relieved heavy barrel, a composite dimensionally stable stock with aluminum bedding blocks to "free float" the barrel (the barrel is not touched by the stock, so that there is no stress placed on it, and each shot is allowed to harmonically resonate without interference), and a good quality scope.
Each rifle is an individual, but I have seen some capable of shooting into a 1/4 "at 100 yards if the operator can do his part. vision headgear through which they look when operating at night. The only purpose for which non-combatants or police would need such equipment would be if they had a serious coyote or other nocturnal predator problem (the distances you cite will be discussed previously, but they are fairly unrealistic).
Scopes do not make rifles accurate. They do not help you shoot better. They help you see better and seeing can be an aid to shooting. But people who do not know how to shoot will not be helped much by any scope, regardless of the cost of it. Some years ago the military was experimenting with proprietary technology in regard to a sighting system, but to the best of my knowledge, the project has been abandoned. The cost was very prohibitive, and somewhere along the line someone figured out that seeing better does not equal shooting better.
Although the military has experimented with other cartridges, you will still find the .308 (7.62mm), in use since Korea, to be the most common "sniper" cartridge. It is a good cartridge, but it has more to do with the fact that we have guns chambered in it, and has more data on the effects of wind and drops that for any other cartridge. The army sniper school requires shots to 800 meters with this rifle. The USMC sniper school requires shots to 1000 meters.
Those are extraordinarily long shots. If the average rifle can shoot into 1/2 "MOA (= 1/2" at 100 yards), that means that it can (all this is predicated on the operator doing his part perfectly, and taking out variables such as wind, etc .) Shoot into 100 at 200 yards, 400 at 800 yards, 500 at 1000 yards. Beyond that distance, drop and velocity become such a factor as to make the round fairly ineffective. The "2 mile" shots are typically made with .50 BMG sniper rifles. These huge cartridges and projectiles are not anti-personnel rounds that are used to take out equipment (communications, truck engines, etc.)
The mission of the military sniper is different from that of the police sniper. The military sniper often looks for targets of opportunity; the police sniper a specific target. The military sniper tries to take his shot from as far away as he can (once he gives away his presence by the shot, the enemy will be looking for him); the police sniper tries to get as close as he can; the military sniper operates in less than ideal conditions; the police sniper usually has a matte to lay on, a flat surface to shoot from, etc. The military sniper is at the mercy of his environment; the police sniper is in control of its environment. A miss by a military sniper usually causes no immediate harm; a miss by a police sniper means that that a hostage will die.
Because of their different operational situations, the military sniper's equipment will trade some accuracy for ruggedness and dependability; Because the police sniper shoots "on his terms" his equipment can be the most accurate available, and ruggedness is not an issue (eg until very recently, USMC snipers had fixed power scopes, because variable power was just too fragile for the field).
To get specifically, unless you are talking about 50 gallon "can," you're not going to hit them from 2 miles away. If that's what you want to do, you'll need a rifle champed in .50 BMG. If you want to do it in the dark, you'll have to get night vision equipment. Obolete Soviet stuff is readily available, but modern manufacture is also, if you're willing to pay for it.