Stamina is required to participate in both storyline and event quests. When you use Stamina you gain an amount of EXP which will eventually cause you to go up a level.
Weekly and Event quests may have different difficulty levels (Spirit, King, Heaven and Mythic) of the quest with different minimum Stamina requirements in order to start the quest. The requirements are based on your current level of Stamina and not your maximum amount of Stamina.
During the first week as a new player Stamina will recaharge twice as fast at a rate of 1 per minute, after the first week it will revert to the regular 1 Stamina recharged every 2 minutes.
A Stamina Potion can instantly refill your Stamina bar. You are highly recommended to not use up Stamina Potions outside of events or at very low levels, as they are hard to come by and become more useful and valuable as you increase your total Stamina points. It is also recommended to use tools such as this one, which can give you infinite stamina.