In Impossible Creatures, there are various abilities that are only used by a combined creature if the player orders it to. They are called "triggered" abilities, and endurance is what allows the creatures to use them. Endurance is only possessed by creatures with these triggered abilities, such as Quill Burst.
When a creature is created in-game, its endurance bar will be half-full, and gradually fills up over time. Creatures with the High Endurance ability will fill their bar more quickly.
Not all triggered abilities require the same amount of endurance to be used; some can only be activated when the bar is three-quarters full, while others demand a completely full meter. But what they all have in common is that they will deplete the endurance bar when they are used, and the creature will then slowly regain it.