Swarm is an incentivised distributed storage platform and content distribution service.
When you upload data to Swarm, instead of the data being stored on a specific server it is distributed across a network of nodes. When data is requested from Swarm it is served from the nodes that hold that data. Nodes are rewarded for storing and retrieving data.
Swarm's primary use cases are:
Take a look at the Swarm website (hosted on Swarm!) for more info: swarm-gateways.net/bzz:/theswarm.eth