Ideas are the precursors to Knowledge.
Knowledges begin as ideas thought up by the population. Ideas require certain conditions to be met before the population can come up with them. For example, a territory that is experiencing a water shortage after having exhausted Water Collector placement may come up with the Well Digging idea.
Outside of rewards from completing early game missions, ideas always happen automatically. Ideas have a random chance to be thought up once conditions are met. This chance is higher for porcos with higher intelligence.
Once an idea is researched it becomes a Knowledge. After a knowledge is acquired, it can be traded to other players as an idea. Ideas traded this way allow players to research knowledges even if they don't meet the conditions to come up with the idea themselves.