If there is a range limit on the Sniper weapons, I have never encountered it. I often put Snipers with Squadsight in Archangel armour and fly them above the LZ and pick off all the enemies that the rest of the squad locates.
All troops have the exact same sight range, which is 17 tiles. If an enemy is within that sight range, it can be shot at by all the regular ranged weapons in the game. However, there may or may not be range penalties, depending on the weapon - shotgun class weapons have a range penalty when the target is far away (> 10 tiles) and sniper class weapons have a range penalty when the target is too close (< 10 tiles).
With Squadsight, the Sniper can fire at any enemy that is within sight range of any allied soldier (in other words, if any other soldier can see the alien). You can tell when the Sniper is using Squadsight as the alien icon will have a yellow crosshairs on it.
However, even with Squadsight, line of sight still applies. If there is any terrain between the Sniper and the alien, then the Sniper cannot see the alien, nor take a Squadsight shot. This prevents a Sniper from outside a UFO from hitting aliens inside a UFO.
The main use of Squadsight is to allow you to take down the aliens without giving them the opportunity to strike back. A single soldier (usually an Assault with Lightning Reflexes) advances one half move towards the aliens, so that they are visible. Then, the Squadsight snipers take their shots. If the aliens are not dead, the Assault retreats back to where it started and you take another turn. All other soldiers are on Overwatch, so that if the aliens move towards your squad to attack, you get free shots on them first.