Comparable to Magic: the Gathering's colors and Cardfight!! Vanguard's clans, the "worlds" are used to categorize the cards into specific groups. Each world has its own artistic themes and playstyles, as well as a unique card frame that represents the world's theme. For example, Dragon World is themed after dragonic creatures, is focused on destroying opponent's monsters, and its card frame resembles fire. Most worlds have at least 3 "main" attributes which can often be used to build decks for them using specific support for them (though those attributes don't necessarily have to be unique to said worlds).
Each world has at least 1 flag card representing them. Those flags generally allow you to use only cards from that specific world and Generic cards. Because of that, it is recommended that your deck only includes cards allowed by your flag card. Some flags such as Dragon Ein and Parade of Hundred Demons allow you to use more than one world's cards by listing a trait that can be shared by various worlds such as an attribute.
With the introduction of Great Fiend, Yamigedo, it is now possible for a non-generic card to not belong to any world. The D Special Series 2 introduces the dual cards, which belong to two different worlds.