A Docking Computer is a ship upgrade which immediately docks the player ship to the target if in range.
The range a ship may dock from varies between stations. Typically for commonwealth stations it will be approximately 5 km. However, for large stations like Terran Solar Power Plants it is possible to dock from much farther away.
A docking computer is not necessary for normal docking, which may be conducted through autopilot or manually requesting permission and piloting to the docking station.
Primarily it is a time saving device for the player because it instantly transports your ship to the docking port. It has in-game implications, for example beating out competing traders to a sale.
It appears that it can dock at 5km away from the port, not the center of the station. So if you're 4.5km from the actual port, but 6km from the station center you can dock. On large stations with many docking ports (e.g. military outposts) instant docking can sometimes be problematic: Your range to target may read under 5km, but you are unable to dock instantly until you find the port that the AI wants you to dock at and get within range of that.