In all of the editions of the core game, up to five players can take control of one of the Axis (Germany or Japan) or Allied (UK, Soviet Union, or USA) nations. The object of the game is to win the war by capturing enough critical territories.
Axis & Allies is not a strictly historical war game, due to its streamlining for ease of play and balancing so that both sides have a chance. For instance, the economic model is simplistic, with each territory producing a number of 'IPCs' (Industrial Production Certificates) good toward the purchase of any unit. Moreover, the game is supposed to start in the spring of 1942, but Japan is immediately in position to attack Hawaii again, while Germany is pressed well into the USSR with an initially superior force. Another significant difference is that the Soviet Union and Japan are opponents at the start of the game; historically, they did not go to war with each other until 1945. If the game were truer to history, the Axis empires would be at their apex at this time, about to be pushed back home - but, of course, that would not be a very competitive match.
Each side's objective is to conquer enemy territory, achieving victory by holding two enemy national capitals; the Axis can also win an economic victory by gaining enough territory, which is more common in practice due to the difficulty of attacking London or Washington.
Each player progresses in order: USSR, Germany, UK, Japan, USA. On each turn, the player declares IPC spending for production and research (which gives a random chance to enhance units). The player then declares combat moves, attacking enemy-held territory. Much of the playtime is spent resolving battles. Each round, all the attacking and defending units get a chance to hit once. A unit scores a hit if the player rolls, with a six-sided die, a number less than or equal to the unit's attack or defense score (e.g., infantry makes a better defender than attacker). Each player gets to choose how to distribute casualties amongst his or her own units and removes them from the battlefield. (This provision makes it practical to bring cheap cannon fodder units along to soak up hits.) The battle continues until either force is destroyed or the attacker withdraws.
After all battles are concluded, the player takes non-combat actions: moving reinforcements, landing aircraft, etc. Then he or she deploys units purchased at the beginning of the turn in territories with an industrial complex and finally collects income from all occupied territories to end the turn.
This cycle continues until, after USA's turn, the victory conditions are met for one side.