As a field of study, it’s the study of two-player games of perfect information (so think chess, not football or poker) in which each player may make countably many moves (you can also look at uncountable-length games but it’s not common). I’ll give you more detail than I would a child :P
Each player takes turns to move. You can encode the moves they make as coming from some set - for example they might just play numbers. The rules of the game are imposed by a winning set, which is a set of countable-length sequences of moves, and we say that player I wins if the infinite sequence of her first move followed by player II’s first move followed by her second move, etc, is in the winning set. Otherwise player II wins. (There are no draws, which technically means chess falls outside the scope of this setup, but it turns out not to be a big deal)
(This allows you to encode what moves are allowed by the rules - you just say that any sequence which contains a move where that player broke a rule is a loss for that player, regardless of what comes afterwards.)
Each winning set defines a different game. The property of determinacy is a property of sets of infinite sequences which says that there is a winning strategy for either player. A strategy is just a function which takes the finite sequence of moves up to that point in the game and tells the player (the player for whom the strategy is) what to do. A winning strategy is one which, if followed, always results in a win for that player.
If we modify the rules of noughts and crosses (tic-tac-toe) so that draws are arbitrarily decided to give a win to player I, we know that this (finite) game has a winning strategy. In fact, any finite game has a winning strategy (or, if there are draws, this means there is a non-losing strategy). The outline of the proof is that if player I does not have a strategy to get to one of the (finitely many) winning states, then we can find a strategy for player II which avoids those winning states. (Remember, winning states are winning for I).
So, which games are determined? Are all games determined? Well, it’s actually easy (through a diagonalisation argument, same as proving uncountability of the reals) that not all infinite (countable-length, that is) games played with natural numbers (as moves) are determined. But you can create a way of categorising the sets of countable sequences of natural numbers (i.e. the possible winning sets) by a kind of complexity. This is the basis of descriptive set theory. It starts with topology: you can define basic open sets in this space as those sets consist of all infinite sequences which share a common finite prefix. Closed sets are the complements of open sets, as usual. But then you can define a hierarchy of complexity where the next level are countable unions of closed sets, then the next level are countable unions of complements of countable unions of complements of open sets. (An introduction to descriptive set theory will say more about this).
It’s quite easy to prove that all open sets and all closed sets in this hierarchy of complexity are determined. It’s a little harder to prove that the second level is determined, and harder still to prove that the third level is. Eventually a guy named Tony Martin (D. A. Martin) proved that all Borel sets in this hierarchy are determined. If you know your analysis, the Borel sets are exactly what you’re thinking: they’re the sets formed by all arbitrary countable unions, intersections and complements of open sets.
The interesting thing about this proof was that it needed a huge amount of set theoretic “power”. Most ordinary mathematics like analysis doesn’t need all the axioms of set theory, but this needed a massive chunk of them. This makes it interesting to set theorists because it tells us something about the relationship between something quite concrete: complexity of sets and strategies for easily-defined games on the one hand, and something quite abstract: the axioms of set theory. This pattern continues higher up: more determinacy can be proved if you assume even stronger axioms, going beyond what is typically included in set theory.
The succinct way of defining the topology (on NxN) is the product topology of the discrete topology(/ies). Maybe that’s the discreteness you’re feeling?
Axiom of Choice is not regarded as a big deal by most set theorists, but it’s interesting when it comes up. The diagonalisation proof that there are undetermined games uses choice to well-order the set of strategies, so it’s actually the other way around: without choice it is consistent (assuming consistency of some other stuff) that all games in this formulation are determined. This is called the Axiom of Determinacy.
The axioms in question are power set and replacement: to prove full Borel determinacy you need to apply the power set axiom infinitely many times (using the replacement axiom). These two axioms are what gives the ZFC axioms their power, really.
Set theorists nowadays I don’t think debate about axioms per se. Set theory nowadays is at once somewhat pluralistic and somewhat settled (paradoxically). I’ll explain a little: set theorists are basically agreed that the ZFC axioms are natural, intuitively “true” (many set theorists would not put scare-quotes there, but I would), powerful enough to do all ordinary mathematics and more, and probably consistent. They also generally agree that many large-cardinal axioms are natural and probably consistent, though there is a wide variation in whether people think they are “true”; there is not nearly so much intuition that such huge objects could exist. This is different than our intuition behind the axiom of infinity, because that axiom is actually needed to do some ordinary mathematics (though you can do without it for a lot!)
The Projective Hierarchy continues the stratification of the Borel Hierarchy even further. If you assume infinitely many woodin cardinals, then you can prove Projective Determinacy. I have heard Tony Martin being quoted as saying that “if Projective Determinacy were found to be inconsistent” (and hence infinitely many Woodin cardinals is inconsistent) “then I’d be having serious doubts about [the axiom of] Replacement.” This gives you a flavour of how people think about the relationship between these concepts.