A double a-side was a single that was released twice, with the same two songs, except that in one version one song was the A-side, and in the other version, the other song was an A-side. In addition to tapping the market of kids who were too cool to listen to a b-side, it also allowed a band to chart twice at the same time. But, saying "a band" is rather non-specific, as I believe that The Beatles were the only band with the chutzpah to pull it off successfully. They released three singles in this manner: We Can Work It Out / Day Tripper, Eleanor Rigby / Yellow Submarine, and Strawberry Fields Forever / Penny Lane. Of these, only one of the versions failed to reach number one. See below for an explanation.
This term has lost all meaning that it might once have had with the advent of the CD and the current incarnation of commericial radio since no modern band has a record-player-owning fanbase large enough to send two seperate versions of the same single to the top of the charts.
But once... there was. In 1967, The Beatles were preparing their response to Pet Sounds, by The Beach Boys, which Paul felt was better than anything The Beatles had done up to that point (which included Revolver, arguably one of their finest albums). Brian Wilson, however, was already trying to match Revolver, an effort which put him into a creative rut from which he would never escape. But I digress.
The original plan for Sgt. Pepper's Lonely Hearts Club Band was that it would be a retrospective of Liverpool, with songs about various things from there. In this vein, John wrote Strawberry Fields Forever, and Paul wrote Penny Lane. But, in early 1967, their record label began wanting more money--and releasing a single was the easiest way. Since The Beatles had nothing else done, they gave the record company these two songs, and decided to release it as a double a-side. The version with Penny Lane as the a-side went to number one, but because of a wide variety of other songs on the charts at that time, Strawberry Fields Forever failed to reach number one, and is thus missing from 1.
In addition, this put the band back to square one on Sgt. Pepper's Lonely Hearts Club Band, because they felt that forcing fans to buy songs twice wasn't fair, and they didn't want to put songs from singles onto their albums (though there are a few instances where it happened). Nowadays, no band in the world has enough good material to release non-album cuts as singles, and even if they did, they couldn't hit number one. It's tragic, really.