A relatively general definition of an infinite, discrete "Gale–Stewart game" is the one used in Martin's proof of Borel determinacy [1]. More information about this type of game can be found in Kechris's Classical Descriptive Set Theory and in Jech's Set Theory.
Definition. We define a game $G(X,P)$ based on a nonempty set $X$ and a set $P \subseteq X^\omega$. The elements of $X$ are the moves of the game. A play of the game is a sequence $r = (r_0, r_1, r_2, \ldots)$. If $r \in P$ then we say that player I wins, otherwise we say that player II wins. Hence every play of the game has a winner. The informal interpretation of this game is as follows: player I chooses $r_0$, then player II chooses $r_1$, then player I chooses $r_2$, etc.
The way that the game is played is formally specified by the definition of a winning strategy. In the game $G(X,P)$, a strategy is just a function $s \colon \left ( \bigcup_{i \in \omega}X^i \right ) \to X$. Intuitively, the value of $s(\tau)$ is the move that should be played if the other player's moves up this this point are listed in the sequence $\tau$.
A winning strategy for player I is a function $f \colon \left ( \bigcup_{i\in\omega} X^{i} \right ) \to X$ such that player I wins any play of the form $$ f(\langle \rangle),\: r_1, f(\langle r_1\rangle),\: r_3, f(\langle r_1, r_3\rangle), \ldots $$
A winning strategy for player II is a function $g \colon \left ( \bigcup_{i\in\omega} X^{i+1} \right ) \to X$ such that player II wins any play of the form $$ r_0, \: g(\langle r_0\rangle),\: r_2,\: g(\langle r_0, r_2\rangle),\: r_4, g(\langle r_0, r_2, r_4 \rangle),\ldots $$
This set of definitions leads to a well-defined, non-paradoxical collection of games. It is impossible for both players to have a winning strategy for the same game, because the strategies can be played against each other. However, it is possible for neither player to have a winning strategy, and in fact ZFC proves that in the specific case $X = \{0,1\}$ there is a set $P \subseteq \{0,1\}^\omega$ for which neither player has a winning strategy.
Donald Martin proved the Borel determinacy theorem in 1975: if $X$ is any nonempty set whatsoever and $P$ is a Borel subset of $X^\omega$ then one of the players has a winning strategy for $G(X,P)$. Here $X$ has the discrete topology and $X^\omega$ has the product topology.
The axiom of determinacy states that if $P$ is any subset of $\{ 0,1 \}^\omega$ then one of the players has a winning strategy in $G(\{0,1\}, P)$. This is inconsistent with ZFC but is consistent with ZF plus dependent choice.
The paper that Andres Caicedo gave above [2] studies an entirely different type of game, which is continuous rather than discrete.