When an effect creates mana, that mana goes into a player’s mana pool. From there, it can be used to pay for spells and abilities. Mana can only be used once, when mana is used to cast a spell it is removed from the mana pool, no longer available for the player to use. Each player's mana pool empties at the end of each step and phase.
Prior to Magic 2010, when mana remains in the mana pool at the end of the player's phase, that player would experience mana burn. Said player subtracts one life for each mana in their mana pool. With the introduction of Magic 2010, mana burn was removed, and mana is now drained at the end of every step. This means that the mana pool empties at the end of the step without any penalty.
One of the few cards which can prevent the mana pool from emptying is Upwelling. Upwelling's text states that "Mana pools don't empty as steps and phases end." This means the mana in each players' pool at the end of each step will carry over to the next step. If you have at the end of your end phase, then you'll have the two green mana at the beginning of your opponent's upkeep.
Cards that Mention the Mana Pool