I just got a MTG card named Liege of the Tangle: card details.
I'm a little confused here. What does that "awakening counter" mean? How long will it remain? Can it be removed from the land?
A counter in Magic: The Gathering is permanent. This means it will stay in the game as long as it doesn't get removed by a spell or ability or the card to which it is attached leaves the battlefield (i.e. by dying, by getting exiled, etc.) This means that every land on which you put an awakening counter will stay a 8/8 green Elemental creature until it leaves the battlefield or something removes the counter.
Nobody can remove the counter without an appropriate spell or other mechanic as mentioned above.
You should be careful how many lands you want to transform. If your opponent has a mass creature destroyer in his hand like Day of Judgment, and he gets to play it, you could lose a lot of lands because they are creatures now as well. I played Liege of the Tangle myself a while ago and I adjusted the amount of lands I transformed based on the deck of my opponent.
The rules clarification says all there is to say. "Awakening counters" are just a certain type of counter that's put on the target lands. They remain on a land as long as that permanent stays on the battlefield, or until an ability removes those counters.