A God Stat is roleplaying gamer slang for a character statistic with a particularly unbalanced effect compared to the other characteristics. It is the opposite of a Dump Stat. God stats are often accidentally created by GMs as a result of their house rules; sometimes, however, they are present in the system RAW. In combat-related games it is typically the stat that governs the number of attacks or actions-per-round the character is allowed.
Dexterity/Agility is the most common god stat in combat-oriented games, often allowing one to hit more frequently and more easily, be hit less, and move faster. It also tends to plug into more non-combat skills than other physical stats, while Strength and Constitution/Endurance/Stamina are much more narrowly-focused on action.
Intelligence is the second most common, since it tends to give out more skill points and offer greater versatility than other mental stats. It's also a popular stat for casting, and casters often out-muscle the competition in terms of both raw firepower and problem-solving ability.
In Mage: The Ascension, the God Stat is literally Arete; it's a measure of how enlightened the character is with their godlike ability to defy the cosmos. It sets the upper-limit to a Mage's magic powers, and high Arete means a Mage can use magic without wasting time with foci or rituals. Maxing it out won't make you overpowered, but one point of Arete is worth more than one point of anything else.
In Mekton, Reflex is the God Stat because without it you simply can't do anything awesome (like be a Mechwarrior, which is what the system is built around) and you will just fall over a lot and get hit all the time (like your average grunt suit pilot - NPCs should have low Reflex.) It's closely followed by Intelligence and Education, which govern your number of skill points. High numbers of skill points allow you to add a fuckton to dice rolls, since they're nearly all opposed checks using Reflex and at least one skill.
In the Fallout series of games, the undisputed god stat is Intelligence. Having a high intelligence gives you the most skill points, so it would be retarded to neglect it. On a side note, playing the (best, original) games as a stupid character with IN <4 can be fun too. Just get a lot of intelligence for normal playthrough and for your 500th char play stupid, it's worth it. This also supports the concept of "Young wanderer taking on the harsh outlands" that the later games tend to have. Even for Fallout 4, which removed skill points, Intelligence is important for leveling faster to get Perk points.
In the Iron Kingdoms RPG, it's by far the Speed stat that again takes the cake - Not only is your Speed stat the total distance your character can move each round, it also feeds into two derived stats: Initiative (which let you go quicker, which, as in all games, is important), and Defense. Defense is what makes you able to dodge attacks, and as most attacks in the game tend to do a lot of damage or even ignore Armor, Defense is the better option. And the traditional enemies of high-defense-low-armor characters in the wargame, templates and fire/corrosion, are rarer and somewhat easier to survive or gain immunity to.