I've seen shooters refer to their weapons as "hitscan weapons". What does this mean?
In general, "hitscan" weapons have no bullet travel time -- when you click the mouse or press the button, whatever's in your crosshair gets hit.
Wikipedia has more info on the history of the term "hitscan": Hitscan
Hitscan is a term is used mainly in computer games. It is a test to find out what can be hit by an in-game weapon (be it a melee weapon, or the projectile emitted by a firearm). Hitscan weapon is often used colloquially to refer to a projectile weapon which uses unmodified hitscan information to dictate whether or not it has hit its target — deploying the weapon calls the hitscan function, and if an object is detected in the path of the projectile, a hit is registered. The effect is that these projectiles travel at infinite speed (a luxury of computers' non-analog simulation of time) and have a linear trajectory — a practical but very crude simulation of a bullet's speed and accuracy. Programmers need not use hitscan data in precisely this way when programming weapons.
Although hitscan weapons behave as described above, they can be given a more 'natural' appearance; for example, the assault rifle in Half-Life 2 calls a hitscan function in the middle of a 'blazing gun' animation, creating a small amount of lag between weapon deployment and the hitting of the target to better approximate real-life ballistics.