A context sensitive button is a button (usually, but it could be a stick position) that has a different function based on where you are and what's going on around you at the time (that is, the context). Some games have a dedicated "action button" that does nothing unless you're in a situation where a special action is available to you; others subborn one of your standard controls when you get into a recognized context. There is usually some sort of on-screen indication that a special action has become available.
This trope is only in play if there are a wide variety of context-sensitive actions connected to a given button, which come up many times during play. If a button changes its use only happens once or very rarely, it's probably not this trope.
This is an Omnipresent Trope in the RTS genre, as well as most computer games that use a mouse and aren't shooters. Actually, it's a common feature mouse use in general; in Windows the right mouse button generally opens a "context menu" that has options relating specifically to whatever you right-clicked on.
Often Action Commands or Press X to Not Die use a Context-Sensitive Button, but they don't have to. For example, if a particular button is always Block, but blocking only works if you do it at the right moment, it's an action command but not context sensitive. Similarly, if you have to hit Block during a Cutscene to block an attack, that's not context sensitive. The button's function didn't change; you just have to time the block properly or use it at an unusual moment.
When there is an In-Universe button that seems to do whatever the plot demands at the moment, that's a Plot-Sensitive Button.
In the 3D games, the A button does just about everything that interacts with the environment and isn't "attack", including jump, climb, roll, push, and pull. The Legend of Zelda Ocarina of Time was actually the first video game to use an on-screen display of what the context-sensitive button would do at that moment.
One specific action button in Beyond Good and Evil allows the player to fight, take pictures, pick up objects, use special keys, jump, talk, climb, kick things and turn pillars. A different action button makes your partner perform some of these actions.
The all-purpose "action" button in Tomb Raider is used to shoot, grab hold of ledges, climb out of pools, or push boulders.
In Full Throttle, the right-click menu always had the same four buttons (hand, foot, mouth, and eyes), but depending on what you were using them on they could have very different effects. For example, using "mouth" normally means "talk to this character", but when used on a hose stuck in a gas tank, it meant "suck" to start syphoning fuel.
First Person Shooter
The "A" button in Gears of War is a catch-all button for the gameplay's duck and cover system. You slide into cover, switch pillars with a swat turn, jump over barriers, roadie run, anything you can think of. In fact, it's so frequently used that players and critics have complained about the button causing the wrong thing to happen because the context changed slightly just before they pushed it.
Portal, Left 4 Dead, and Half-Life have the "Use" key (default: E) which will do various things based on context. In Portal, it normally picks things up, but will also push the small red buttons on pedestals and open unlocked doors.
Portal 2 uses the same all-purpose use/grab key, but the tutorial has a subversion: when Wheatley asks Chell to speak, the game prompts the player with "Press [jump button] to speak". But when you do, all she does is jump.
The trope namerConkers Bad Fur Day actually has platforms that the game refers to as "Context Sensitive Buttons", which have the letter B on them. When B is pressed while standing on one, it does whatever the game requires at that point, from providing Conker with dynamite to getting him drunk.
In Sonic Adventure, there's one specific button that can make you attack, pick stuff up, put stuff down, or use the floating help TVs in the Chao Gardens. This can be a problem.
Sometimes, you can try to pick up your Chao and end up attacking it.
Sometimes, you try and put down an animal in the aforementioned Gardens and end up in a help menu because you were next to one of the TVs.
Then a Chao comes along and takes the animal out of your hands, changing in ways you didn't intend it to.
Role Playing Game
In Chronicles of Inotia: Children of Carnia, everything aside from movement, pausing, switching characters and skills gets bound to a single button. This includes both attacking and talking, so the game can become Unwinnable by Mistake if you're cornered by monsters while standing close enough to converse with somebody; your attack button turns into Talk and there's no way to fight your way out.
Both Kingdom Hearts: Chain of Memories and Kingdom Hearts II do this with the triangle button, which is specifically for "Reaction Commands". The attack button also performs various 'interact with this thing' functions when you are in the right position to do so.
Dream Drop Distance also has Free Flow Action, where pressing the Y button will cause Sora/Riku to do a variety of acrobatic maneuvers, depending on the environment, from grinding rails to latching on to enemies.
The Elder Scrolls V: Skyrim has a particularly annoying example of this. The control to place an item in a container is the same as to pick up EVERYTHING in the container, depending on whether your are taking or adding things.
A Tamagotchi is a virtual pet that's controlled by only three context-sensitive buttons.
Stealth Based Game
Under the Hitman control scheme, pressing the "Use" key may cause the player character to perform any one of literally hundreds of actions, depending on context. Walk up to certain NPCs with your hands empty and "Use" will start a conversation with them. Sneaking up on them from behind with a pistol drawn or looking down on them through the roof hatch of an elevator with your garrotte equipped will yield far less benevolent options.
Batman: Arkham Asylum makes heavy use of this trope. At appropriate times, almost any of your controls can change to a context-sensitive command. Each button generally sticks to a theme — 'jump' makes you do something related to movement, 'punch' does a takedown of some sort, and so on — but not always.
The Oddworld series uses these starting in Munch's Oddyssey. The 'jump' button is also used to drink from a vending machine, sit down in your wheelchair, pick things up, etc.
Wide Open Sandbox
The Assassin's Creed games use this trope quite a bit; the four buttons on the controller generally correspond to "do something with your head", "do something with your feet", "do something with your left hand", and "do something with your right hand". What exactly each one does depends on whether you're standing still, running, riding a horse, climbing, standing close to somebody, and so on.
Touchscreens on smartphones are big context sensitive buttons, functioning depending on the programming - how you use the screen changes according to which app and what is on screen.