Knowledge is the resource name given to how much credit you've earned towards your next technological unlock. Unlike in some other games, where you have to produce research locations and give them direction towards what you want to eventually unlock, in AI War you simply go gain some knowledge and store it somewhere until you're ready to use it.
Once used Knowledge is lost, like metal or crystal resources. It can only be replenished by using labs vessels on friendly planets that haven't had all of their Knowledge absorbed, or via Knowledge Raiding which is stealing knowledge from AI controlled worlds. Knowledge Raiding can be a risky proposition and thus the most common form of gaining Knowledge is simply researching on friendly or neutral planets with supply.
Each planet contains 3000 knowledge that can be absorbed by science labs. Science labs are built from a command station in the econ tab. Science Lab 1 is your basic lab, Science Lab 2 cloaks and absorbs more knowledge/second, and Science Lab 3 is primarily used for Knowledge Raids. All labs are available from the beginning of the game.
Applying knowledge to gain new technology requires a science lab of some kind available. Open the science lab (Hotkey: S) and you will see a display of available technologies. Hovering over each of the lit buttons will show you the item you can open and the knowledge cost. If you hover over a darkened button, you'll see a Prerequesites item there that will tell you what else you need to open to be able to open that item as well. There are no complex science trees in AI War. Under most circumstances to open higher marks of something you simply need to open them in order.