Sniper Scope Sway

The key features of a Sniper Rifle is that it is accurate at long ranges, usually with a telescopic scope to be able to properly target at range and the power to take targets down in very few shots. Ideally one shot, one kill -- especially if properly aimed.

Because the view through the scope is magnified many times, the slightest movement of the gun will cause the sniper's view to shift accordingly; even when the sniper holds their gun as perfectly still as their arms allow, videogames and movies will still show the view swaying in synch with the character's breathing (and often in a subtle, sidweays figure-eight pattern). This serves to emphasize that sniping is more difficult than merely point and shoot, and snipers may compensate this in a number of ways, such as lying prone against the ground, using a tripod to brace the gun, or holding their breath while preparing to take the shot.

For videogames in particular this also serves to counterbalance to the rifle's power and range against other weapons. Because the sniper is usually too far away for their targets to effectively return fire (save for the occasional Sniper Duel), making long-range kills more difficult with scope sway (in addition to other limits, like the rifle's slow rate of fire and limited ammunition magazines) helps prevent the rifle from becoming a Game Breaker in competitive multiplayer.

Examples of Sniper Scope Sway include:

Metal Gear Solid features sniper rifles in every game, as well as being able to either take pentazemin (or a cigarette) to relax and reduce the tremble when scoping.
Call of Duty and its various sequels - the games after the original have scope sway, which can be lessened by crouching or going prone, and let you hold your breath to steady your aim for a few seconds, after which the sway will be even worse until you get your breath back. You can also use perks to extend how long you can hold your breath. The same is true of the thermal scope, while the ACOG scope has a lower zoom level and less sway, but you can't hold your breath. Changed around in Modern Warfare 3, where all types of sights have some degree of idle sway, though actual sniper scopes make it the most obvious and are still the only ones you can hold your breath with.
Silent Scope is fairly unique in that since the arcade machine has an actual sniper rifle to hold, complete with a zoomed in screen to allow you to use the scope, any sway and bob to your view is the result of your actual movements. As a result the real world techniques of relaxing, holding your breath and so on apply, but since the rifle is mounted and you're typically not aiming at targets that are too distant, you don't need to worry about it as much as real snipers.
The Deus Ex games allowed you to level up your rifle skill, which among other bonuses would result in the scope swaying less, with the crosshairs perfectly steady at the highest level. Rather than sway related to breath, the crosshairs do a random walk.
Similarly No One Lives Forever 2 also allowed you to upgrade your sniper skills with the same effect.
Day Of Defeat scopes would sway off target.
In Hitman, it depends on your rifle. If you are using a fully upgraded W2000, there is little sway, though it only gets a single shot(though one can still get high precision with a magazine). Also, sway is affected if you move and by the length of time you hold the rifle after sighting. Sway can be eliminated by canceling the sighting and adjusting your position to the next target before sighting again.
Max Payne: Pretty standard. Hold your breath and all movement ceases. Move and the sway becomes much worse, crouch and it lessens. The sniper shot bullet cam can get pretty annoying.
The Legend of Zelda: The Wind Waker has no sniper scope, but Link's bow sways anyway.
Sniper rifles in Borderlands have different amounts of sway as a hidden attribute of sorts: it can be reduced by crouching and certain class skills.
Mass Effect 1 features ridiculous amounts of Scope Swaying for any player who has not invested significant points into the Sniper Rifle skill. However, when the skill is maxed out, the rifle is steady as a rock, and the active talent for the skill eliminates it entirely for one shot at any level. Mass Effect 2 features noticeably less right from the start.
Ghost Hunter - The sniper rifle, with zoomable scope, features this. It can be steadied some by kneeling to fire.
Splinter Cell games require you to hold your breath to get a steady shot when using a sniper scope.
In Worms 3D, this happens when the player is using the shotgun since, like in other Worms games, the shotgun works more like a two-shot sniper rifle.
Perfect Dark has this feature for all scoped guns, but it's especially bad on the sniper rifle. To compensate, the player can crouch while firing for greater stability.
Fallout 3 and Fallout: New Vegas have sway on all scoped weapons, especially sniper rifles. The higher your skill for that weapon type the less sway you will have to deal with, and you can also crouch to make the sway less severe.
The Ballistic Weapons mod for Unreal Tournament 2004 features sway for every weapon, including sniper rifles - move and jump around a lot and the sway gets worse, stand still or crouch and your shots are pinpoint accurate, so long as you don't fire too often or get shot by someone else.
Red Orchestra: Ostfront 41-45 has sway for all weapons. More sway when low on stamina from sprinting or from aiming for too long, but almost none if the weapon is rested on a surface. The sequel removed the stamina based sway, but added controlled breathing.