Browser-based, serverless apps are what Ethereum is all about. You run a daemon on your computer that maintains a blockchain, messaging system (Whisper), and content storage system (Swarm). That daemon exposes an RPC interface so local processes can get data from it. The most common result is an HTML, CSS, and Javascript bundle that is a full-fledged application without a server, because its backend is peer-to-peer.

Web apps aren't a requirement—you could build a traditional desktop app that talks to the Ethereum daemon as well. But using the application runtime that everyone already has installed lowers the barrier to entry, so it's what most successful projects will choose.