Worms: With quick fingers it is possible to switch to the rocket launcher and the mini gun at the same time, causing the minigun firing code to fire rockets. There is usually a significant chunk of the level missing afterwards.
Colonization has a bug on the trading screen which lets you sell resources that have been forbidden through a Customs House. Naturally, most players don't bother installing the patch to fix this bug, as it makes the game a good deal easier towards the end.
In fact, the open-source clone FreeCol reproduced this as an optional rule.
The original Civilization had the Settlers bug: if you ordered a Settler unit to perform some multi-turn task (like irrigating or mining a square), it'd usually stay inactive until the task is completed, which e.g. in case of mining a mountain would take up to 9 turns. However, if you canceled its current action and then ordered it to do it again, the engine would interpret it as if the Settler already spent a turn doing it and continue from where it "stopped". This trick could be repeated until the task was complete. Sure, it's a lot of manual clicking but it allowed you to fully irrigate/mine the area around AND build roads your city in a few turns, which is crucial at early stages when every Settler counts.
You could also put this same Settler on a transport and use this bug to build a road and then a railroad in the middle of the ocean. A ship could carry cargo halfway across the map if you have an ocean railroad in place for it. You'd have to end a unit's turn when it's loaded into the ship, but other than that you can go VERY far.
In addition, save games did not keep track of which units had already used their turns, so long as you still had at least one unit who had a move left. You could play through an entire turn, save the game, load the save file, and play another turn without letting any other nation get to play.
Civ I gave the AI single blanket aggressiveness ratings. The Democracy government made your citizens unhappy with war, so it gave a modifier of -2 to the rating. This worked pretty well at making them less likely to start trouble, except that Gandhi's default aggressiveness was 1, and would underflow to 255, so as soon as he got democracy he would turn into a bloodthirsty maniac that threw nukes around with reckless abandon. (By way of comparison, the most warlike leaders were Montezuma and Genghis Khan, with aggressiveness of 10.) This was so memorable that later games in the series give Gandhi an abnormally high priority on the construction of nuclear weapons.
In Civilization II: Test of Time, there were game modes with multiple "worlds", which mapped to each other 1-to-1. Normally, only certain units could move between certain worlds in certain circumstances, which was enforced by unselecting the active unit when the player manually flipped between maps. However, if the player controlled a city on the map a unit was on and another on the map they wanted the unit to get to, then they could select the unit they wanted to move, enter the city screen of any city they controlled on the starting map, use the buttons in the city screen to cycle over to a city on the target map, then exit the city screen. The unit would still be selected and could thus be given movement orders, so as long as the square they would be moving onto on the target map was passable to them (all worlds used land/sea designations in some way), they would glitch onto the new map mid-step.
In Civilization IV, getting a Great Person would allow you to expend that unit and put research towards a new technology, initiate a Golden Age, build a unique building, boost your culture, etc. Also added in this game was the ability to give multiple orders at once to a unit, by holding shift while clicking their abilities. It was soon discovered that holding shift and clicking on the abilities of the Great Person did not cause them to be deleted. One game advanced from the Ancient World to the modern Industrial Era through the continual brilliance of Moses. Sadly, this bug was fixed with the expansion packs.
In Civ 5, on the higher difficulties, you can leech the gold boost that computer players get by exploiting them with a good army; take over one or two cities, then offer one city back and peace for a whole lot of gold. Go to war 10 turns later, repeat. Usually, conquering a civ only gives you a small fraction of their gold and the rest is lost (for game balance and other reasons), but this way, it can be extracted from weaker civs. This one isn't entirely a Game Breaker since it requires some work and might even make a little sense, besides the computer being far too slow to learn what you're doing (if it ever does). Before the 2011 patches, there were a lot of other ways for players to mess around with numbers in unintended ways.
There is also another glitch/exploit that utilizes trade system. Basically, starting a war cancels all agreements you've made with someone You could - however - trade long term investment (strategic resources) for one-time payment. And if you start a war very next turn, the game automatically broke the deal and give you all your resources back - which means that in the end you get both the money and your goods. Oh, and for added irony you could use that money to fund an army that beat said player.
Native villages/ancient ruins in Civ IV and V could also bring...interesting results. In IV, there was no upper cap on what tech you could get from them - so it's entirely possible to find perfectly working moderntechnology (up to freakin nuclear fission) in some tribal village on the remote corner of the map. In V, they fixed that mistake by limiting possible techs to the classical era... but, at the same time, give you a chance to upgrade your units. Again, there is no limit on how high this could get - so, if you are lucky, your cities could be guarded by machineguns and riflemen. In 1000 BC.
X-COM allowed your soldiers to throw grenades through the ceiling. This was incredibly helpful when there was a Muton on the roof.
X-COM had another gem of a bug: Whatever you set the difficulty to, the game played on the 'Beginner' difficulty. This bug went unknown for years, because the game was THAT damned hard. And it gets better: when MicroProse was throwing together the mission pack sequel Terror from the Deep, they solicited feedback from players. A big chunk of players who responded were hardcore strat gamers who complained that the game's higher difficulty levels weren't hard enough. MicroProse's response? Make TFTD's Beginner difficulty equivalent to what Superhuman was supposed to be in UFO Defense.
There's also the pathfinding bug in UFO Defense for the blaster launcher, where the majority of the time the enemy units would launch it directly above themselves, even if there was a roof above them, leading to many times in indoor maps where there would be numerous random explosions going on out of your line of sight for, from your perspective, no reason.
XCOM: Enemy Unknown, in its expansion Enemy Within, has the "carryover bug", where an upgrade to the MEC's weapons' damage is carried over to subsequent games, where it can be applied again to further boost the damage.
In Heroes of Might and Magic III, there was a campaign (Rise of the Necromancer) in which you are given two very powerful artefacts; the Armour of the Damned (that makes every enemy weak, slow and useless in battle) and the Cloak of the Undead King (which turned the rather useless skeletons gained from Necromancy into the much more useful, ranged Liches). These allowed you to quickly build up an invincible army of liches that would number in the hundreds by the end of each map. You only get one of each set, but the campaign would also take copies (as well as the component pieces). If you used the Dungeon's Artifact Merchants to gather the components again, you could go through the rest of the campaign with multiple Heroes made unstoppable with these artifacts.
The Apple II game Taipan! had the player as the captain of a trading/fighting ship in the Far East. In Hong Kong, one of the ports you could visit, was a moneylender (loanshark) with a ruinous interest rate. Unfortunately for Elder Brother Wu, he didn't understand the concept of negative numbers, so you could borrow 100 and pay him back 200, at which point he'd start paying you interest at his usurious rate.
Galactic Civilizations: The diplomacy screen lets you trade basically anything in the game: from single ships and technologies to treaties to entire solar systems. And of course money. Different techs and buildings boost your diplomatic abilities, effectively giving you the upper hand in negotiations, thus letting you acquire more ships for money, or more tech for your money, or - that's right - more money for your money. Things don't stop there, though. You can make contracts where you and the AI pay each other a certain amount for x number of rounds. For some reason this makes both parties richer every round, effectively granting you infinite cash. Another good thing is that the AI just can't handle that much money, utilizing only a fraction of what it has available, while you crank out disgustingly strong fleets in no time at all. Too bad GalCiv doesn't have a multiplayer mode.
Super Robot Wars Judgment had a bug where, if you applied an item to a unit that would be lost due to game events (such as Mazinger Z or Layzner), whatever they would replace (Mazinkaiser and Super Layzner/Layzner Mk-II) would get those bonuses even though the item would be returned to you.
Super Robot Wars Z2 had some interface tweaks between the first and second parts. One of the tweaks opened up a bug that allows you to have all of your party members take a turn at using the Sub Orders system, instead of the normal limit of twenty. While this can be used to gather enormous amounts of money very quickly, a less game-breaking application is to gather kills and PP on the units that you don't normally deploy on a given run. Another bug allows you to select and then deselect the carryover option when starting a new game to stack the carryover bonus money, allowing you to gain enough money to max out all of your units before the game even begins.
One of the patches for Master of Orion 2 introduced a very odd AI quirk. Every so often an AI controlled race might notice that your science and military make you significantly stronger than they are. So it would decide you are a threat and declare war on you. A few turns later the AI will take a look at you again and realise they can't possibly win a war against you, so they will ask for peace - often throwing in one of their planets as an incentive. This can happen repeatedly - so the enemy literally hands over their empire to you one planet at a time.
Masters of Orion 2 also features a bug that lets your ships become invincible. The best stealth field module makes your ships untargetable and immune to damage if you don't attack that turn. There's also another module that lets your ships act twice during a single turn. Combine the two and you can use your first action to attack, then do nothing on the second one (which the game treats as you not attacking that turn) to become untargetable to the enemy.
In the first Master of Orion the AI cannot manage more then one or two attacks at a time. This allows the player to win simply by hitting his enemy at a faster rate.
Endless Space used to have the exact same AI bug as Master of Orion 2 above, where the AI would sometimes repeatedly declare war on you, only to immediately ask for peace. Fitting as the game is pretty much a Spiritual Successor to Master of Orion 2. While the bug was fixed by improving the AI and adding a minimum time you have to wait before you can declare a war again after agreeing to a ceasefire, another very similar AI quirk was still possible the last time I checked. If you have a large amount of extra minerals or trade goods, and one of the AI factions really dislikes you, you can use them to get free stuff. When they declare war on you, offer a ceasefire and hand them enough goods in return that they will not only agree, but also throw in a planet or some technology. When the ceasefire ends the AI, who still hates you, will declare war again, cancelling all trade routes you had with them and returning any minerals/trade goods you agreed to trade to them back to you. Repeat the process for as long as necessary for free stuff, then crush them with the armada they've helped to finance.
There was also a bug in the early version of the game that allowed you to eliminate another faction by literally buying their empire from under them. In practice if you could do this you either had enough money to win by economic victory anyway, or the faction had so few planets they were insignificant, but it was still pretty funny. Later patches made it so that the AI will never sell you all of their planets.
There's a glitch in the seventhFire Emblem game that lets you control your enemies. By placing a mine on the ground, waiting for an enemy to step on it, and resetting the game while the HP bar decreases, you are able to move your enemies just like you would your own units. It's only for one turn, but that's enough for you to make everyone drop their weapons (or hand them to you) if you wish.
The "Uberspear" that gives major boosts to Vaida's stats when she appears as a boss is not a bug. Being able to steal it off of her and enjoy the stat boosts yourself is.
In Fire Emblem: The Sacred Stones, that enemy control glitch (which can now be activated by normal combat damage as long as it's the enemy's turn) still lets you make all the enemies hand their weapons and equipment to your team. Including, say, claws, teeth, and eyes, which the game treats as weapons. And if you then equip eyes, you can train any character, no matter their class, in dark magic. You can also give rare items to a gorgon egg, which will mark its whole inventory except the first item as "items which will be dropped when I die", wait for it to hatch, and kill it, and when you do all the items will be fully recharged, even the unique items which were specifically designed to be unrechargable.
As well, you can recharge elixirs by letting an enemy thief steal one and then kill him. Same thief trick also serves to recharge Myrrh's Dragonstone.
And while their status as "Glitches" or "Bugs" is highly debatable (As both are results of correctly working systems in the game), there's at least 2 tricks that allow your to pair The Hero Seliph with his half-sister, and resident Mysterious Waif, Princess Julia. (First involves forming a particular square of units and not saving for 2 and a half chapters; the other involves having Seliph and Julia glued to each other until she's captured, then have Seliph wait next to a Brainwashed and Crazy Julia on the final chapter).
And finally, there's a possibility that even the plot itself can fall victim to a glitch, or exactly the unbridled WRATH of the Random Number God— it's possible for Ethlyn and Quan to kill the Thracian Knights in Chapter 5... the one where, plot-wise, they're supposed to be murdered.
In Fire Emblem: Radiant Dawn, an enemy Bishop with a Rescue staff can place an enemy on top of a pitfall trap in Chapter 3-11. This allows a unit with Pass to move through the square unimpeded, as well as creating a very silly situation where a unit can engage an enemy and then fall into a pit as soon as they try to pass through the recently occupied space. More importantly, you can take advantage of this by placing your own flying units atop one of these spaces (and you recruit a LOT that chapter) and then moving your non-fliers past the pitfalls. The game will not allow two units to occupy the same space; therefore, it will not stop them as they go over the pitfalls.
In Romance of the Three Kingdoms IV: Wall of Fire, one can transform a number of soldiers into a nigh-infinite number of soldiers by assigning them to a general's command and then declining to go to war, removing the general and reassigning him, and repeating as desired.
In Xenonauts, if you leave an injured soldier in your base to recuperate, it will take many days. However, in earlier versions of the game, sending him or her into battle again and using a simple medikit would fix the problem instantly. This was eventually fixed before the final release.
Earlier versions of Advance Wars 2: Black Hole Rising, much like the Fire Emblem glitch above, had a means to take control of the enemy army through careful abuse of soft resets and the map maker. If you pulled it off, all you had to do was choose "Yield" from the options menu to win the match instantly.