Claiming a system can be valuable itself, as it secures the system with a clear indication of rules you can set. Expanding your faction either from a military perspective or through structures that can help your faction grow is also an important aim, and is required to undertake certain faction activities such as fabrication or research.
Some structures only require a prefab to build, such as the buoy, but most structures will require several structure components to construct. Before any of this can be attempted, however, you will first need to secure a ship with the construction role such as the Civilian Guild Engineer, Serf or Worker class ship. All of these require the Construction Craft skill to use which is within the negative range of the sensitivity factor (the Worker however only requires Construction Craft 1). If you are yet to realise this skill you should complete the basic flight training mission from the missions window.
The Worker ship has limited space and the structure components will need to be in the hold of the ship to perform construction. However, if you are commanding a ship with bays (like the Battlecruiser) you can dock a construction role ship, then the structure components only need to be within the mothership. To stow a ship in a bay, find the parked ship and be present at the same location. Then access the parked ships section from the command menu and choose the contextual option to stow the ship.
Regardless of which ship you choose the aforementioned buoy only needs the buoy prefab structure component to build, this prefab can be found at most systems that trade parts as it's a low level component. With this component in the hold of a construction role ship (or using the docking method above) you can build the buoy in any system that is not claimed. To perform the construction, open the engineering window and choose the option to build, next to the blueprint of the buoy. The buoy requires no power to operate so can therefore be used to claim systems in remote regions.
For structures that require several components you will either need to purchase these components or fabricate them at a factory.
Most structures also require power to operate. Power is shared across the entire system so power structures can be built on any body within a system. The most basic power structure is the solar collector which can be placed in orbit.