The purpose of iron sights is often for a game to feel more realistic, as the player is aiming through the actual sights of a weapon. In early FPS games like Doom, players did not need to reload or truly aim weapons. Since then, most FPS games feature reloading. The same is happening with iron sights. GoldenEye 007 on the N64 was the earliest FPS to feature both reloading and some form of aim-down-sights, inspired by light-gun rail shooter Virtua Cop. At first, very few games used this to its full extent. Even in a game such as the soldier simulation Operation Flashpoint, you had a regular crosshair that worked just fine to aim with, but you could precision aim with iron-sights.
Vietcong was one of the first games that took the step that if you aren't firing through your iron sights, you are firing from your hip, with much reduced accuracy, and now more games are using this as a standard feature, like the Call of Duty franchise. Games like Red Orchestra and America's Army also use iron sights to a great extent to impact and change gameplay. Games like Gears of War do not use iron sights because of their third person perspective, but the idea is exactly the same, you have one button to shoot with and one to aim with, if you don't aim, you don't have a crosshair and suffer from reduced accuracy.
Impact on Gameplay
Iron sights may be used to impact gameplay in various ways. Often you are slowed down and have a reduced field of vision when aiming down the sights, which often makes you an easier target as you line up your sights with your target. Iron sights are most often used in more realistic games such as America's Army or Operation Flashpoint. But there are cases where iron sights are used but the game is still frantic and very fast, like Call of Duty 4.