Similar to building a colony ship to take an uninhabited planet under your control, you can also build 'Constructor' ships to build Starbases. There are five different types of Starbases, each of which have things in common and distinguishing features. All starbases allow you to equip weapons and sensors to them to protect them from attack.
Influence starbases help to expand your realm of cultural influence, which can sway nearby planets or asteroid mining bases to your control.
Economic starbases increase the production of the planets within it's radius, research, currency, and industry.
Military starbases can increase your own ship's defense, attack, and speed within their radius, as well as decreasing enemy ship's speed.
Mining starbases are built on special resource nodes spread throughout the galaxy, and increase your overall production of these resources galaxy-wide.
Ascension starbases are a special type of mining starbase only availible in the Twilight of the Arnor expansion (remember, Endless Universe allows you to play this or Dread Lords). These starbases can only be built on ascension crystals (max of 4 per map) and get you closer towards an ascension victory. They cannot be upgraded at all.
Terror Stars, again only avaiblible in the Twilight of Arnor expansion, are basically 'Death Stars', mobile conquest bases that can destroy planets. They are only accessible by some races, and only in the endgame.
These starbases play a strong role in how much power you have, and each one allows you to push towards the non-standard victory types.