A Lightning Rifle is a directed-energy weapon used by the human resistance.
The rifle is large and cumbersome, requiring two hands to wield effectively. After a brief charging sequence, the weapon emits an intense burst of lightning able to destroy a Sentinel with one or two well-placed shots. It is mainly used by the resistance as a last resort when Sentinels breach the hull of a hovercraft. The weapon also features an attached monitor and various other electronics, evidently used in target acquisition.
Despite designed for use against machines, the weapon is also easily able to kill a human target, demonstrated by Cypher when he kills Dozer and injures Tank in the original Matrix film. A shot with the weapon was the equivalent to being struck by lightning or coming in contact with a live wire, though more indirect shots would only burn and injure organic targets. Direct hits cauterized or incinerated organic targets though, if a target were indirectly hit, they would not be fatally burned by the electricity. Tank subsequently uses the rifle to kill Cypher, having been left on the floor beside his presumed-dead body.
The lightning rifle does not appear to be intended for intense combat, and is limited to a short range and slow recharge time. Some of the resistance infantry were equipped with lightning rifles during the defense of Zion, although their shortcomings made them relatively ineffective against the massive Sentinel swarms out in the open. However, they performed better when the machines were forced into narrow passages and tunnels. During the Battle of the Dock, Zee uses one to kill a Sentinel with two direct shots and save the Kid, allowing him to shoot the chains to open the gate.
These weapons don't have a formal name in the scripts. They're alternatively called "energy rifles", "lightning rifles", and "plasma rifles".