Most, but not all, buildings run on some kind of power. This goes for resource extractors, manufacturing/processing buildings, agricultural buildings, and research buildings. Each planet must provides its own power; it is not shared between planets. Your power sources can be found under the lightning bolt icon; available options differ from planet to planet.
Most extractors, factories and processors run on Fuel or Energy. Fuel comes from refined Methane or Oil, and is your mainstay fuel source when you start. Some buildings run directly on Fuel, while others run on Energy, which can be generated using Fuel by way of Small Generators and Thermal Plants. These power sources provide low amounts of energy, but are sufficient for your early years.
Later, as your power needs increase and you unlock Nuclear Powerplants and Fusion Reactors, Fuel becomes less important and Hydrogen becomes your main source of power. Hydrogen can be extracted from gas giants, meaning Orpheus will be your main source for a while (with an extraction rate of 8x), and you can make up the remainder by extracting it from Water using Electrolyzers.
You can expect an eventual bottleneck until you capture Posirion, your first truly Hydrogen-heavy plane (51.30x extraction rate). Phorun (31.30x) and Mermorra (15.4x) are also Hydrogen-rich, and with the three of them, you'll have your power needs covered for quite a while.
Some planets, however, will find their power needs best suited in other ways. Planets close to a star can thrive on Solar Power, which needs neither Fuel nor Hydrogen. A Solar Central on Ishtar Gate (2.8 AU from a star) might give 6 energy/second, on Promision (1 AU) it would give 50 energy/second, and Lone Nassaus (0.1 AU) would get 5,000 energy/second.
Ocean planets can build Hydroelectric Plants and Pressurized Water Reactors, which require water and, in the latter case, Uranium.
Other power options include Battery Power Planets, which can be useful at times, but are generally impractical and inefficient compared to other power sources.