There are two types of structures in Battledawn - Production/Resources (which produce Food, Metal, Oil and Energy) and Strategic/Military (which helps your armys & defences). Structures can not be destroyed or captured.
A full list of structures can be found in the Structure List.
Building structures is easy. Once you have enough metal and oil, you can build it, the picture of the structure you want will not have a red cross on it. For example, the Metal Mine: Level 5 needs 5000 metal and 2500 oil and once you've started building, it will take 24 hours before you gain the benefit from the structure, while it is building there will be a countdown timer over top of the structure's image.
You may build one structure from each row at a time, and as long as you can afford each structure you can have 1 building in every single row if you wish. The building "time" occurs on both structures independently (I.E. There is no "Build Queue") so the structures will all build simultaneously.
The Structures at the left side of a row (lower level structures) are cheaper to build then the ones on the right side of a row (higher level structures) e.g. Energy Reactor is 500 and 500 but Energy Reactor: Level 5 is 5000 and 2500 . Generally, one gains a bigger benefit from higher level structures than lower level ones, to make up for the larger cost and building time.