Memoir '44 is Richard Borg's landmark game of World War II tactics, notable both for excellent production quality, including a plethora of little plastic solders, armor, and artillery, and for a game system that, while fairly complex by Eurogame standards, is nonetheless extraordinarily simple by board wargame standards, while retaining the strategic complexity of far more rules-complex board wargames.
Over the years, it's been extended and expanded; the initial game dealt with the Normandy landings and combat in France in 1944, but a huge number of additional maps and scenarios have been released, including ones set on the Eastern and African fronts, and in the Pacific theater.
Although played on a hex grid, it has very much the feeling of a miniatures game; each unit can take four hits before elimination, which you represent by removing individual soldiers or vehicles from a group of four. There is no combat results table; rather, when a unit fires, you roll between one and three dice (the number determined by range). Each die has a grenade symbol on one side (inflicts a hit), infantry symbols on two sides and an armor symbol on one side (inflicts a hit only if the target this type of unit), a star (a miss), and a flag (target retreats if it took any damage).
Each player has a hand of command cards; typically, they let you move and attack with a handful of units, but only in one of three areas of the map (left, center, and right).
As typical with wargames, terrain, defensive entrenchments (wire, sandbags, bunkers and so on) and effective use of combined arms tactics are critical to effective play.
Eliminating an enemy unit scores a victory point; typically, there are also objectives on the map that score one or both players victory points when occupied. The first player to reach some number of victory points wins.
Memoir '44 is almost always a tense game, with players agonizing over which card to play, cursing or crowing over their luck as the dice do or do not go their way, and more often than not, the victor uncertain down to the final moments of play.
You have to assume that the designers of Armies and Allies or Army Attack have never played this game; it makes their combat systems look totally inane. As they are, of course.
Days of Wonder have recently released an online version of the game, available for PC, Mac, and Linux; while you don't have the tactile pleasure of moving lots of soldiers and tanks around playing online, you do have access to a huge number of scenarios, and can play either solo or against live opponents. And there are enough fans of the game that you almost always can find someone to play with.
The business model is an unusual one; you get 50 "gold ingots" for free, and 2-4 are consumed each time you play a scenario. So you'll get a fair bit of free play, with no restrictions, by downloading the game. You can, of course, buy more. I'm sceptical about this business model; I used something very like it for a game on Sony Station a decade ago, and it was not at all successful. But of course, they can always try something else if it doesn't work out for them.
Still, Memoir '44 is a superb game, a wargame far more accessible to a widespread audience than typical board wargames; and the online implementation is nicely done, and worth your time to check out.