IT WAS a week or so ago that the battle for the world championship of chess reached that climactic stage at which the American public begins to show about as much interest as it ever does in the game by sitting down and making a serious effort to get it straight: Who's playing whom? It doesn't make it any easier when the names are Karpov and Kasparov. Now, which one is it that's the model communist, defending champion, darling of the establishment? Which is the brash challenger from Baku with the daring style of play and an irreverent attitude toward the powers that be?

The challenger, and winner, as a good many Americans can tell you by now, was Gary Kasparov. Over the weekend he concluded a brutal grinding-down of the champion, Anatoly Karpov, by winning the 24th and final game of their series in Moscow. Mr. Kasparov did so with an audacious strategy aimed at winning, even though all he had to do was gain a draw to ensure himself the championship.

Mr. Kasparov's victory had his fans in the crowd cheering and chanting his first name, which was something of a breach of decorum, but certainly not the first in the prolonged and bitter struggle between these two chess players. They began hammering at one another in September of 1984. Mr. Karpov moved far ahead in that championship match, but then began to falter under the relentless pursuit of the challenger, whose training regimen includes long-distance running. Many thought Mr. Karpov was on the verge of physical and mental collapse, and when the head of the international chess federation stepped in to stop the match, Mr. Kasparov and many of his supporters were infuriated, believing the chess establishment was attempting to save one of its own.

The hard feelings carried over to the new match, which began in September of this year and ended Saturday. Now Mr. Karpov's side says he may demand another match soon, although he looked a little peaked to us in those final days. Mr. Kasparov says the two should hold off for a time. The new champion seems likely to create more interest in chess among Americans than anything since Bobby Fischer's ascendancy. "If Kasparov remains the champion, he will definitely influence the style of many young players," grandmaster Lubomir Kavalek told The Post's Joseph McLellan. "His attacking style is clear and understandable; if he hits you, you know you are being hit." Tune in for the rematch; faint hearts may prefer to stick with NFL football.