Expanding is basically enlarging your base to close or remote locations. Expanding can be used to secure Ore Nodes, Tech Structures, or to set up a foward base. It is common to expand your MCV, build some structures, and then pack up and move somewhere else.
Allies
The Allies can expand with their Prospecters. The Prospector deploys into the Command Hub, which gives a larger amount of ground control than the Soviet Sputnik, but takes more time to deploy. However, each Command Hub must be teched up individually.
Soviets
The Soviets expand with their amphibious Sputnik vehicles. They unpack into Outposts, which give less ground control than Command Hubs, but unpack much more quickly.
Empire of the Rising Sun
The Empire has the easist way of expanding thanks to their Nanocores, which can deploy almost anywhere on the battlefield. Nanocores are excellent for taking Ore Nodes.