There are four primary types of stations in EVE Online: NPC stations, conquerable stations, and player-owned stations and outposts.
NPC stations are found all over high-sec and low-sec space and are an important part of the game for everyone as they serve as trading hubs and a base of operations. NPC stations are generally huge and offer a variety of useful services in addition to allowing docking for all but the very largest of capital ships.
Conquerable stations can be found in many 0.0 regions. Similar to regular NPC stations, they are not built by players and can not be destroyed, but can be taken over and run by player corporations.
Player-owned starbases (or simply POS) are stations that are built by players or corporations. They come in several sizes and styles and can be equipped to perform a variety of tasks. For instance, they can be configured for refining, moon harvesting, or production. They can also be equipped with potent defense systems that require a capital ship to take down.
Outposts are large player-owned stations built in 0.0 by alliances, and require a significant investment of resources and effort. They are the closest thing players can build to the large NPC stations.
It should be noted that none of the player-owned stations are anywhere near as large or cool as the big NPC stations found in the game. Nevertheless, some of them look decent enough and are a big status symbol. Smaller POS can be built by very small corporations, with a few even being built in wormhole space.
If your goal is to build a POS, you can certainly do that given enough effort and patience. Building an outpost, however, requires an organized effort and isn't sometime a small corp can accomplish on its own.