Language games are subsets of languages as a whole as you use them in everyday life - that's it. You can have multiple language games within a single overarching natural language (such as English), and English itself can be seen as one massive language game. A language game needs players - that is, people to use the vocabulary for the purposes for which it is intended.
An example of a language game might be the theology of a specific religion (as is appropriate for these forums). Catholic theology, for example, has developed a rich language game over the years, with jargon such as transubstantiation, immaculate conception, true god from true god, begotten not made, salvation, grace, faith, works, you name it.
While these terms are usable in one sense in everyday English, the language game of Catholic theology is also in a sense more self-contained than English as a whole - you need to be a player to understand how these terms are used in everyday life, and how Catholics debate about religious doctrine using them. They all have special meanings for the players, and that meaning simply is the way they're used - or rather, there is no such thing as "meaning" proper, but only the way people use words.
Wittgenstein thinks that problems arise when you take vocabulary from a language game and export them into another language game. So, for instance, if we were to take the Catholic concept of the soul and import it into the language game of Western science, then we might ask a question like "is the soul measurable? Does it have mass? Where does it go, physically, upon death?" But this is a violation of the language game - in the "home" language game in which the word "soul" is used, these types of questions as applied to it
don't even make sense - the scientist is looking for an answer where there is none, and in the process falsely thinking that he is either discovering paradoxes or refuting the claims of Catholicism.
So, all philosophical problems that seem to be head-scratchers - for example, "how do I know that there exists a real world corresponding to my mental representations?" are actually instances of inappropriately crossing language games, and are problems to be dissolved by clarifying the use of language, rather than solved on their own terms. That is because the terms don't make sense - since languages are just actions we perform and the way we use words, we have smooth sailing if we grease our language games to use them appropriately and deal with the world efficiently.