Spirecraft - Hard, Broken Golems - Hard, Botnet Golem - Hard, and Fallen Spire are all game settings that can cause 'standard' exowaves to appear. These exowaves charge up slowly, and a warning appears when the reach 50% charged. The strength and charge time varies based on game time, AIP, or other in-game events.
Certain in-game events may trigger exowaves. The Fallen Spire campaign is an example where you may encounter this situation. These types of exowaves do not give any warning, but they are triggered directly by something the human player does.
Exotic and Vicious Exotic are AI types that replace that AI player's normal waves with exowaves. Strength is computed directly from how strong the normal wave would have been.
The Vengeful AI type will send small exowaves at the player every time the player destroys a structure belonging to the AI. The importance of the structure increases the strength of the exowave. For example, killing a guard post will trigger a small wave. Killing a command station will trigger a larger one.
In addition, destroying any AI type's Core Guard Posts will trigger a small exowave in retaliation.
All exowaves appear the same way: As a number of independent groups of ships, appearing at randomly selected Warp Gates through AI held territory. Generally, they will appear 1-2 hops from human territory. Each group of Exowave ships will move as a group (based on the speed of the 'Leader' ship), and share the same targets.
Exowaves will generally include a smaller number of ships (each group will cap out around 100-200 ships), but the ships will be much stronger than those that appear in normal waves. This, plus their mobility and unusally aggressive behavior, makes many players consider them to be more dangerous than normal waves.