What is a game? What is play? Do these intersect?
While there is no general consensus on the answer to the questions, Avedon and Sutton-Smith  posit the following definitions:
Games are an exercise of voluntary control systems in which there is an opposition between forces, confined by a procedure and rules in order to produce a disequilibrial outcome.
Play is an exercise of voluntary control systems.
Their definition of play is fairly vacuous and only makes sense in context of the definition of games, but their definition of games is quite interesting. Many others who define games insist that games are also non-productive. This means that they do not result in production of valuable goods. That requirement is meant to increase the delineation between labor and gaming. Still, the details of the Avedon and Sutton definition are quite telling. It is a requirement that in a game there is a disequilibrial outcome. This means that in the conflict of oppositional forces there is a winner and a loser. It is also necessary that a game operate procedurally. This is quite different from the situation that occurs when one watches a film or reads a novel. The experience of gaming is necessarily oriented toward the goal of winning under this definition. This structures the experience of the player. Though a game may also incorporate meaningful content at a variety of levels (such as simulation, narrative, interaction), the player is typically not oriented toward considering these aspects as the primary purpose in engaging in the gaming experience. We see now that the definition of play includes childhood activities such as playing house, and computer simulation games such as the Maxis's Sim City series, while the definition of game does not.
All this does not mean that games cannot provide meaningful experiences. It is important to separate the medium from the delivery method and social conventions surrounding it. For example, the video medium is capable of providing incredibly rich and possibly transformative experiences, but the conventions of network television preclude this. One way to analyze this phenomenon in the realm of games is to realize that games operate on several levels simultaneously.
Eric Zimmerman  likes to describe games as functioning as a sets of rules, social interactions, and cultural phenomena. The set of rules are the constraints the define the way the game is played. They delineate what is considered "legal" within the game. The social interactions define the way that the game is actually played and the manner in which people interact while playing the game. For example, while playing soccer it is perfectly legal to sit on the grass, play with ones toes, and bellow like an elephant, but the social interaction protocols surrounding soccer disapprove of such activity. In the board game Monopoly, players typically trade and barter game cards (that represent real-estate) according to social protocols not specified by the game rules, indeed many consider this to be the fun of the game. Games also function at a cultural level in that communities form around them. They have also potentially have an effect on aspects of society at large. The Pokemon game phenomenon completely transformed large segments of the population of young people's play habits. People gather for social events such as playing bridge or dominoes, and the cultural status of chess Grandmaster is often quite exalted.
Now it is possible to relax any of the requirements of the definition of games provided above and still have something that is "game-like." A better way to look at this is to say that the techniques of procedural, interaction, and cultural design in gaming can be recruited into the design of other cultural forms such as storytelling, simulation play, fine arts, theatre, or dance. Even within the realm of gaming it is important to realize that popular forms such as multi-player first person shooters, massively multi-play roleplaying games, or real-time strategy games are not the only viable types of gaming. As with television, the dominant distribution methods and business models define the most popular forms of the medium. But there are still video artists and documentary filmmakers. There are many hobbyist game designers producing shareware applications, and there are many interaction designers producing game-like online experiences using tools such as Macromedia's Flash and Director/Shockwave. The company Gamelab is a good example of this. Outside the realm of gaming, there is a long history of procedural interactive narrative. The international writers' group Oulipo has been constructed witty texts using procedural methods since the 1960s. Hypertext fiction has been an influential, if not overwhelmingly popular, form since the 1980's. Brenda Laurel, Chris Crawford, Joseph Bates, Don Marinelli & Randy Pausch, and Michael Mateas are just a few of the people currently working in this area. You may be interested to search online and investigate the work of any of these artists, researchers, and designers.
 Elliott M. Avedon & Brian Sutton-Smith (ed.): The Study of Games , New York: John Wiley & Sons, Inc., 1971.
 Katie Salen, Eric Zimmerman. Rules of Play : Game Design Fundamentals , MIT Press, 2003.