Its four sides bear the letters nun-gimmel-hey-shin, which some people take as the initials of nes gadol hayah sham, “a great miracle happened there”, an allusion to the Chanukah story (the Israeli version replaces the last letter with pey, short for po, “here”).
When the top is spun, the numerical value of the four letters indicate the score, respectively 50, 3, 5 and 300. Some interpret them as denoting nichts, nothing; gib, give; halb, half, and shtell, put in (an alternative view is that the gimmel is ganz, everything; according others the shin is shalem, pay).
Of course we all know that Judaism sternly opposes addictive gambling and regards it as personally and socially destructive. I have heard it said that on Chanukah the gambling somehow conveys a theological lesson in that the letters on the dreidel add up to 358, which is the numerical value of Mashi’ach, Messiah.
A better argument in favour of the game is that a little fun is a valuable thing. If the only gambling we do is when the Chanukah lights are burning it may prevent us from becoming compulsive gamblers, in the same way that a little wine on Shabbat and festivals may have ensured that Jews were able to remain more or less free from alcoholism.
Since playing with the dreidel needs money, maybe that is why we have the custom of giving Chanukah gelt (Chanukah money) to children, though a better reason for that custom is that it reminds us of Maccabean independence, which was commemorated by means of minting coinage.