Golem is an open, peer-to-peer network. In a traditional network, computers can act either a server or a client. When a user wants some work done, they use their local machine as a client to send messages to a server, typically located in a remote datacenter. While datacenters do everything to guarantee they are online as close to 100% of the time as possible, they usually cost quite a lot.
In a peer-to-peer network, each computer is at the same time client and a server, so each computer is equal and thus a peer. In a peer-to-peer context all of the computers forming a network are usually called nodes, while nodes connected directly to a particular machine are called its peers. This is why we call computers running the Golem app nodes on the Golem network.
If you want to be a Provider, Golem node will perform computations for money, acting as server. If you want to be a Requestor, Golem node will orchestrate computations for you, acting as client.