In early FPS games, crosshairs didn't exist mostly because all enemies were placed in one horizontal plane. Aiming was controlled by directional keys rather than the mouse. Even if enemies were above or below the first person perspective, the player had to just line them up on his plane of movement, as was the case for Doom and Duke Nukem 3D. At the time, the mechanic of aiming with crosshairs in first-person view was largely limited to arcade light-gun shooters, such as Gun Buster (also an early FPS) and Virtua Cop, with the use of a light-gun.
Eventually, PC titles such as Marathon and Quake revolutionized the FPS genre by combining mouse aiming with mouse camera control and placing a reticle in the middle of the screen to make precise pointing in three dimensions possible, controller aim and auto aim reticles followed suit.
However, later games that aspired to be realistic, removed crosshairs again. Instead, the player has to guess where his shots would approximatively land. This effect has been used creatively to only restrict certain weapons, like sniper rifles in Counter Strike or in Tactical Ops. Moreover sometimes the lack of reticule signifies that it is next to impossible to hit anything when shooting a gun from the hip, because the cone of fire without a scope is artificially increased.
In games that just remove cross hairs without also impeding accuracy, the absence of a reticule can be circumvented by painting a dot -using a marker- in the middle of your screen.
In modern games, reticles have been partially or completely replaced by the use of iron sights, scopes, red dots, and particularly laser sights.