Engagement is one of the core mechanics of Apex Predator that is used by all races and card types. Some actions take multiple turns to complete, and engagement is the system which is used to keep track of those actions.
At the beginning of the game, all cards start in the "ready" position, which means that they are oriented vertically such that the owner may read the text on the card. Most actions in the game require you to "engage" a card, which means to rotate it 90° clockwise. Some actions require you to engage a card two or three times, which indicate that it should be turned 180° or 270° clockwise, respectively. During the prep phase, the player can "ready" each card once, which means to rotate each card 90° counterclockwise (or keep it in the ready position if it is already there).
Most actions in the build phase — including gathering, building structures, and researching upgrades can only be initiated with cards that are in the ready position. Similarly, during the move phase, only cards which are in the ready position are eligible to move. As a result of these two rules, it is almost never possible for a card to perform a build phase action and move during the same turn.
During the combat phase, each card's engagement and combat classes are the two key attributes that determine which nearby cards it will be eligible to attack. The combat classes page contains a more detailed description of this, but for a simple rule of thumb: ready cards can attack anything, cards engaged once have some restrictions on what they can attack, and cards engaged twice or thrice cannot attack.