The computer game based on the board game was developed and published by Hasbro Interactive in 1998. As far as I can tell, it is only available for Windows 9x.
The game is, simply, the board game. It boasts AI players (of dubious intelligence), various multiplayer options (IPX network, Internet, and hotseat, the last of which is the most recommended way to play), support for both the second and third edition rules (or some things from the first), and a few "house rules" that can be turned on or off. Players of the original board game will probably be familiar with the second edition rules, while players of the two expansion games (Axis and Allies: Europe and Axis and Allies: Pacific) will probably be familiar with the third edition rules. A rule of thumb, if you're not sure which ruleset you've played with, is that in the second edition battleships are sunk in a single hit, while in the third they take two hits.
I'm not going to go into detail about the many rules options available, but you can play either the plain second or third edition rules, or you can turn on any of the individual changes from the second to the third editions. There are also about a dozen little other rules you can turn on or off, none of which are official board game rules, but can spice things up.
The AI is stupid. It's very very stupid. The UK, for example, will almost always build a factory on India on its first turn. A smart player of Japan would thus be able to take this factory, and march all over Asia. Also, a smart German player can unfailingly take the Soviet capitol on the second turn. If you've played the game before, you'll understand just how stupid this is.
So far, this game actually sounds fairly cool. This is because I have yet to make light of this game's weakness, which is this: it's buggy. It's extremely buggy. The release version was so buggy, in fact, that I'm not even going to acknowledge its existence, and I'm only going to talk about the patched version. It is fairly obvious that Hasbro's motivation was not to make a solid and enjoyable game.
One bug, a fatal one, goes as follows: suppose you wish to land troops on an empty territory from a transport, and the water square in front of the land contains enemy ships. In the game rules, you first move all the ships (the fighting ships or planes you use to kill the enemy ships guarding the water, plus the transports) to the water square and the units on the transports to the land square as a combat movement. Next you resolve the combat, with the condition that the troops only make it to land if the transports survived the naval combat. The actual claiming of the empty territory is considered to be part of the "combat" phase.
Where the game gets confused has to do with the fact that it's expecting an amphibious assault against the land territory (because you fought a naval battle to get there), except there's no one there to fight. Thus, it waits at the "choose combat" screen forever, because it expects you to choose to fight in the territory in question when there is in fact no fighting to be done. The only way to recover is to load the Autosave file, which it saves at the end of every round. This can lead to repeating a great deal of the game. Saving often is recommended.
The game has many other, less fatal quirks. It has been known to hang or crash the computer on occasion, though, I admit, mostly if the patch isn't applied. The save game feature can lead to odd inconsistencies if it's saved at certain times, but this is easily avoided.
This game does have a few advantages over playing the actual board game. Mainly, it avoids the lengthy setup time of the board game, as the placement of pieces is done by the computer and the need to fiddle with tiny plastic pieces is lifted. Also, no disputes over the rules arise because the computer handles all of that rather forcefully (indeed, if you want to get someone into the board game, getting them to play this first might be a good way to get them familiar with the rules rather than forcing them to read the lengthy rulebook). It's actually possible to pop off a whole game in the space of an hour with this PC version, something not quite possible with the board game.
On balance, though, playing with the real board is more enjoyable, and much more social. This is not a replacement for the board game, but can be a good companion to it. If only there were as many freeware A&A games as there are freeware Risk games...