Buying land can be done via the Land > Buy Land screen, it can be bought for ingame money … however a tile can only be bought if one is for sale. There are 10,000 tiles per realm but because the game world would be cluttered if all were up for sale at once there are never normally more than a couple of tiles for sale at any one time.
Tiles can be put up for sale by players who have previously bought a tile, and also I sometimes put a series of tiles up for sale to mark special events such as holidays, and the games release anniversary.
Use the Land > Buy Land screen to look for tiles that are currently up for sale, only tiles that show up in color are available on this screen and if you click on an available tile you can choose to buy it should you have enough money.
Note: Buying tiles is expensive and not intended for low level players.
Also there are 2 bought extras the Orb of Terraforming and the Genesis Orb which can be bought for real money via the File > Buy Extras > Amazing Items screen. They allow you to claim any tile in the game which is not yet claimed and not yet protected as a system tile so this means that even if a tile is not up for sale you can still claim it using the extra.
As a tile owner you can protect your tile making it so that only people with permits can build on your tile, and you choose who you give permits out to, you can also give out relocation permits which allow people to move existing buildings they might own to your tile. You can also make your tile a town tile, which makes it show up on the World > maps screen.
Other options available to a Tile owner include the ability to add special items to the tile, NPCs and customise existing buildings by giving them new names and pictures allowing you to form a tile.
You no longer show up invisible on a tile you own.