Volumes
On-disk files in a Container are ephemeral, which presents some problems for non-trivial applications when running in Containers. First, when a Container crashes, kubelet will restart it, but the files will be lost - the Container starts with a clean state. Second, when running Containers together in a Pod it is often necessary to share files between those Containers. The Kubernetes Volume abstraction solves both of these problems.
Docker also has a concept of volumes, though it is somewhat looser and less managed. In Docker, a volume is simply a directory on disk or in another Container.
A Kubernetes volume, on the other hand, has an explicit lifetime - the same as the Pod that encloses it. Consequently, a volume outlives any Containers that run within the Pod, and data is preserved across Container restarts. Of course, when a Pod ceases to exist, the volume will cease to exist, too. Perhaps more importantly than this, Kubernetes supports many types of volumes, and a Pod can use any number of them simultaneously.