I found the term "run’n’gun" in an article about video games. What does it mean in that context?
Exactly what the name implies, the opposite of sniping, sneaking or camping. Shooting while running — remaining in motion to be a harder target, confusing opponents by brash (suicidal?) tactic, spraying bullets while in motion.
Many games introduce accuracy penalties (on top of naturally increased difficulty) for this tactic, encouraging stopping, aiming precisely and using terrain for cover, while in others it's essential, as immobile targets become overly easy to hit. This is frequently where "dynamism" of a gameplay comes from — classic Quake 3 Arena would have you killed within seconds if you stop, while in Stalker: Shadow of Chernobyl "run and gun" is bound to get you killed as you run into a hard-to-see anomaly or get swarmed by enemies, while sniping, stealth and careful dodging is the way.
An official use would be in Fallout: New Vegas perk:
You should also note a specific, related term kiting, denoting running backwards while shooting a chasing opponent (who is only equipped with a melee or short range weapon). It's a common necessity in many games and requires a good knowledge of the layout of the game, as you don't see where you're going.
Little late for this reply but I noticed no one included that a Run N' Gun is a real training/recreational event in which you carry your firearms/gear over distant areas with cover of varying stances and targets. Its the best training. Second only to live fire exercises (your fellow soldiers fire in preset locations/situations in a controlled setting but you can die accidently) and combat itself. Nutnfancys youtube channel covers Run N' Gun events.