Category 1: Game Play, or the formal play of games
This is the focused kind of play that occurs when one or more players plays a game, whether it is a board game, card game, sport, computer game, etc.

Category 2: Ludic activities, or informal play
This category includes all of those non-game behaviors that we also think of as “playing:” dogs chasing each other, two college students tossing a frisbee back and forth, a circle of children playing ring-around-the-rosy, etc. Ludic activities are quite similar to games, but generally less formalized.

Category 3: Being playful, or being in a play state of mind
This broad category includes all of the ways we can “be playful” in the context of other activities. Being in a play state of mind does not necessarily mean that you are playing – but rather that you are injecting a spirit of play into some other action. For example, it is one thing to insult a friend’s appearance, but it is another thing entirely if the insult is delivered playfully.

by Eric Zimmerman, “Narrative, Interactivity, Play, and Games: Four Naughty Concepts in Need of Discipline”