Claiming with Griefprevention
You can claim your land with a golden shovel in your hand and right clicking the two corners of the area you want to claim (or do /claim).
The more land you want to claim, the longer you must play on the server, as blocks accumulate per hour you play (you must be actively playing (not afk) to receive blocks. The minimum claim is 10 × 10. Even if you think you will want to claim a bigger area, at least start by claiming a small area first, as you can always expand your claim later! To expand or shrink your claim, you can drag the glow stone edges in the same way as you claim land. Select a glow stone corner first!
To share your land claim, you need to stand inside the area that you have claimed and then type /trust <username>.
It is your responsibility when you have someone trusted. If this person steals/griefs, it's your own fault. Players are responsible for their own Claim.
Claiming with Towny
Claiming with towny is a bit different, as it costs money to claim and maintain the land. But, you have much more control over the land through settings, and can even set a spawn point for your town for others to teleport to. This option is best for people who want to have an official town or nation, rather than a singular base.
To start, use /town new <name> to create a town with the specified name. The price of a new town is $3000. The chunk you are standing in will automatically be claimed when you do this. To expand your town claim, simply walk into the chunk you want to claim, and use /t claim. Upkeep per chunk per real life day is $100.