These days, they are called "trading cards" and are sold in packs of assorted numbers sacrilegiously free of actual bubblegum. The industry is a bizarre racket, complete with bonus cards in variant editions with or without holograms and artificially under-printed "rare" cards in any series that can be held off the market and sold at above-the-odds cost. Sports and comics-related cards are still popular, but there are underground artists out there cranking out serial killer collectibles, among much other strange detritus.
In Britain, where bubblegum wasn't much of an option during rationing, cards with series titles like Birds of the British Isles were issued with cigarettes or tea, but in America it was bubblegum, that commodity used almost as currency by GIs in search of a good time, that accompanied the cards. The big player in the field is Topps, which has been producing series of baseball cards for many years, yielding collectable ephemera even more obscurely valuable than rare postage stamps. Topps put out, and rather quickly withdrew, the original Mars Attacks! series.
Oddly, the inspiration was not the Fifties and Sixties craze for science- fiction paranoia that powers the film but the then-recent success of a line of cards celebrating the centennial of the American Civil War. That series had historical respectability, but mostly featured extremely gory battle scenes. It was reasoned that the series was popular with kids - not because they wanted the history lesson but because of the splatter. And the Mars Attacks! series, drawn by Bob Powell and Norm Saunders and masterminded by then 21-year-old Len Brown, delivered even more violence, with the big-brained aliens zapping sundry American icons (a particularly prized item is Card No 36, Destroying a Dog) and giant insects munching down on victims (which Tim Burton strangely omits).
The 55-card series was slipped out as if it was along the lines of Topps' other hits - US Presidents, Railroad Trains, Flags of the World - and was confiscated by horrified parents and teachers almost immediately, then pulled off the market due to complaints. Paradoxically, this suppression made the series amazingly collectable and valuable. A complete, original set will cost you $2,000