[MODERATOR] Please move this to chemistry.
DaveC's answer is correct except for one significant detail. Sodium chloride is not one of the species that gets converted to hypochlorous acid. Chloride ions (Cl in -1 oxidation state) are quite stable .. much more so than other forms of chlorine, such as hypochlorous acid (Cl in +1 oxidatation state). Like the OP surmised, chlorine is a highly electronegative element ... once it gets that extra electron, it's not giving it back without a fight. In sciencey term, you need to add energy to convert chloride ions into just about any other chlorine containing compound.
So, to answer the OP's question .. salt water pools are NOT chlorinated simply due to the addition of sodium chloride. There is also an electrolysis cell which uses electrical energy to oxidize the chlorine atoms (i.e. to steal back that extra electron) so that they form hypochlorous acid. One supposed benefit of this is that the chlorine levels in the pool are much lower than for a normal chlorinated fresh water pool. My guess is that the tradeoff is that they require a lot more regular checking and maintenance to make sure that proper sanitizing is occurring ... however, that is only a guess.
[EDIT] To answer DaveC's question about what happens to all the sodium ... the answer is, probably not much. Sodium is much harder to reduce (i.e. give an electron back to) than chlorine is to oxidize. So, what most likely happens is that sodium stays in solution as an ion, while what gets reduced in the other half of the electrolysis cell is H+ ions. To satisfy charge conservation in an electrochemical reaction, you always need to have the same number of electrons involved in both the oxidation and reduction half-reactions. So, for each chloride ion that loses an electron in oxidation, another species must gain an electron. Based on experience, I am almost certain that it will be H+ ions getting reduced to form hydrogen gas, but I can't say for sure, since I have not studied all the equilibria involved with a salt-water pool.