A Decentralized Application is an application that runs on a blockchain, a good example of a DApp is Ethereum. DApps can be written in any code language, if you want to get a little technical, If an App = frontend + server, then since smart contracts are what runs on blockchains, then DApp = frontend + smart contracts, this concept goes hand and hand with Decentralized Autonomus Organizations as these DApps will be the core services of some DAOs if not the whole DAO's platform. These Decentralized Applications are paving the way for the future of this new industry, and how the community responds matters greatly.
DApps on Ethereum are some of the most interesting and also head scratching applications you'll see so far, but are exactly what is needed to establish some sort of grounds on what to even develop and design as a standard. These DApps will be what the whole world will be watching and talking about over the next couple years, as they are a direct link to the development of cryptocurrency's future. How the users of these applications respond to the applications and the cryptocurrency itself will ultimately define them while shaping altcoin future.
DApps like DopeRaider, Bancor, Decentraland CryptoKitties, have been created and much more, what comes of all of this innovation and ummm playfulness is really an innovative live test on how the public will utilize and innovate these core concepts and mesh them with their own interests, ideas and passions, I'm pretty sure what we will get will be the Internet 2.0 entirely.