When you place a claim, there's a visible bounding box to show you the regions (you can show / hide this for yours and others).
Claim size is yet to be determined but see Jim's post above as it may change.
You can filter out what you see in the world by type, keyword, rating, etc.
You can tag your claim with a title/description and others will see that when they reach your claim ie: "Pyksel's Castle of Doom".
You can rate other claims.
You can package up your claim and sell it or transfer it.
Claims will have an upkeep and if you fail to pay the upkeep, your content is saved but your claim will be released.
Anytime a wipe (that isn't a character wipe) will occur, all of your stuff will be packaged up for you so you don't lose it.
Fast travel will be available to your claim once you've gone through enough progression. Fast travel will also be available across the continent/worlds but you'll have to explore the world.
The expansion of your claim (adding on to an additional claim) horizontally & vertically (up) will be possible while vertically (down) will be somewhat limited. People can NOT build above/below your claim. They can mine under it but can't claim/build under/over it.
After alpha, you won't start with a claim flag as you'll have to go out in the world and earn it. (help to prevent griefing by creating multiple accounts)
They are "charging" for alpha because they want serious testers to come in and help since the game is so rough but you don't have to pay to play.
There will be an NDA during alpha.
Will there be danger & risk in Landmark? Yes! They showed a video of a man eating plant for example.
Day 1 of alpha and what you can expect:
Building
Exploration
Claims
Some basic form of crafting
*Combat will not be in at day 1 of alpha