The landfall ability triggers whenever a land enters the battlefield under your control for any reason. It triggers whenever you play a land, as well as whenever a spell or ability (such as Rampant Growth) causes you to put a land onto the battlefield under your control. It will even trigger when a spell or ability causes another player to put a land onto the battlefield under your control (as can happen with Yavimaya Dryad's ability, for example).

When a land enters the battlefield under your control, each landfall ability of the permanents you control will trigger. You can put them on the stack in any order. The last ability you put on the stack will be the first one that resolves.

Eternity Vessel's landfall ability cares only about how many charge counters are on it, not what its controller's life total was as it entered the battlefield. The number of charge counters may be changed by other spells and abilities.

For your life total to become the number of charge counters on Eternity Vessel, you actually gain or lose the necessary amount of life. For example, if your life total is 4 when Eternity Vessel's landfall ability resolves and Eternity Vessel has 12 charge counters on it, the ability will cause you to gain 8 life. Other cards that interact with life gain or life loss will interact with this effect accordingly.

In a Two-Headed Giant game, your life total is the same as your team's life total, so Eternity Vessel enters the battlefield with a number of charge counters on it equal to your team's life total. When its landfall ability resolves, it causes its controller to lose or gain the appropriate amount of life, and the team's life total is adjusted by the amount of life the player gains or loses as a result of this ability. Suppose Eternity Vessel has 15 charge counters on it when its landfall ability resolves, and its controller's team has 11 life. That player's life total is 11, which becomes 15, for a net gain of 4 life. The team's life total becomes 15.