Additional bases will provide your empire with more economy, besides giving you more places to build fleets and research.
To build an additional base you will need to use an Outpost Ship, so you need to produce it first.
After producing the Outpost Ship you will need to move it to an empty Astro (planet, moon or asteroid) and select the Build Base tab in this fleet menu.
The first additional base will have a setup cost of 50 credits, and each additional base after that will have a setup cost 2x of the last one.
After building the 2nd base you can advance to the next mission.
Mission:
Build a 2nd Base (0/1)
Requirements:
Construct a level 8 Shipyards (7/8)
Produce 1 Outpost Ship (0/1)
http://delta.astroempires.com/tutorial.aspx