To claim your own land, simply build something there. Once you have built ten blocks of your own (and nobody else has built in the same place) you will have claimed the zone for yourself.
Once you own the land, only you can dig or build there. This prevents other people damaging what you have built, or stealing from your mines.
The owner of the land you are standing on is displayed at the top of the screen. It will say something like:
Zone(3,5) Private Land(725)indicating this land is owned by player number 725, or it may say something like:
Zone(0,0) Public Land(0)showing that this is public land. If you own the land, it will say something like:
Zone(9,2) My Land(612)
It is possible to give land away to a friend. Maybe they gave it to you so you could build something for them. First, go to the place you want to give away, then use the command /owner followed by their name or number. Be careful to give the land to the right person, because there's no way to claim it back again!
You can also disown some land if you don't want it any more. Use the command /owner followed by the number zero, and the land will become public again, so anyone can dig or build there.
Some trusted players have special privileges, and can change the ownership of land they do not own. They have the power to take land away from players that have been naughty, and reward good players by giving land to them. They use the same /owner commands shown above, and can also just use the /owner command alone to claim the land for themselves.