A verb is a physical interaction with a controller (or piece, dice, etc) as a part of taking an action in a game. It is to select, move, click, press, talk or do something else in the real world that then translates to a game-world effect.
When the ratio between action and verb is 1:1, they are essentially the same thing. A player pulling the right trigger on a joypad controller to shoot a gun is an example of a 1:1 ratio.
Many actions are composed of multiple verbs. Even a simple action like moving a chess piece involves picking up the piece, moving it legally and then placing it, which is three verbs. Ordering multiple units to move in StarCraft involves moving the mouse, selecting the units and then the location on the map to which they should move. That can involve anywhere from three to thirty verbs.
Sometimes one verb produces many actions. Holding down a trigger in shooting games usually invokes automatic fire, which can mean that through one verb (‘hold’), the player may kill many enemies (actions). Similarly, holding the accelerator in a driving game is just one verb, but can lead to a stream of actions as vehicle position and momentum changes relative to the environment and other vehicles.
While actions are infinite in their variety, in practise there are only a small number of verbs that almost all games use, like ‘press’, ‘select’, ‘drag’, ‘drop’, ‘hold’, ‘push’ and so on. As new controllers (such as WiiMotes or touch screens) have appeared in recent years, the available list of verbs has slightly expanded (‘wave’, ‘pinch’, ‘swipe’ etc).
Verbs are also sometimes called inputs, impulses, interactions or game atoms.
Verb is a widely-used term in game design theory, but its usages are often confusing. Chris Crawford is generally held to be the originator of the term as a way to describe the interaction with software usefully. However some other writers (for example: Raph Koster) talk of verbs in a more sweeping sense, suggesting that the pieces on a chess board are verbs because they have the potential to cause change in the game world.
What Games Are separates verbs from actions to get away from this kind of meta-usage (with no offence to Koster) because it is hard to visualise, and therefore understand.