A loop is how a game determines the results of an action and feeds that result back to a player. Players are usually the initiators of loops but, in videogames especially, characters may also initiate their own loops. An enemy taking a shot at you is trying to win, just as you are.
Loops all have a the same structure: The player performs an action, the results of that action are determined according to rules, states of different resources in the environment are updated according to the outcome. The loop closes with one of two results, win or loss, and the player may have the option to take further actions.
Some loops are closed immediately, where others are hours or even days long. Loops are also differently arranged. In some games like Bejeweled they are strictly sequential. In others they are concurrent or interweaved. In some turn-based boardgames that separate move and resolution phases, the player sometimes makes a number of actions that all stack up, waiting for a set of loops to all be resolved at the end. In others they operate in harmony, or even simultaneously, between players. Some games let the player initiate loops at their own pace, while others operate at either the game’s pace or that of other players.
Loops are subject to both synchrony and temporany, leading to four classes of loop: single (asynchronous, atemporal), multiple (synchronous, contemporal), serial (synchronous, atemporal) and parallel (asynchronous, contemporal). In successful games, loops cluster to form repeatable dynamics, and so become fascinating in the long term. In poor games, this clustering either does not happen or happens badly.
The psychology of the open loop is very important to understanding why games are engaging. The play brain remembers and is agitated by actions left unfinished, and the reside more presently in memory than past occurrences. The analogy of waiters remembering the status of all of their tables is apt in this regard. Each is an open loop, a question to be answered and learned from.
In this essay by Dan Cook, he describes loops as 'skill atoms', their components, how they link and a system for mapping loops as a chemistry of your game design. The phrasing is deliberately intended to treat the subject of game design as a science, elaborating on how atoms can form 'molecules' (which What Games Are refers to as 'dynamics') and overall build a game out notationally and logically. It also slants very heavily toward understanding games in terms of skill acquisition and mastery.
While the scientific phrasing is perhaps too obscure for many readers (and is why What Games Are doesn't use it), Cook's work in the whole area is fantastic and well worth a deep read.