What is this purple claw thing (Demon Altar) and what can I do with it?

They are Demon Altars which are not monsters - hence you cannot get the tooltip by hovering over them.

You use them to craft some 'boss' summoning materials Slime Crown, Suspicious Looking Eye and Worm Food, as well as creating the stronger pre-hard mode weapon Night's Edge.

Demon Altars can only be destroyed with Pwnhammer or Hamdrax after Wall of Flesh is destroyed. Trying to destroy Demonic Altar before defeating Wall of Flesh, you will take half your current health in damage minus armor, no matter the tool. Pwnhammer is acquired by defeating Wall of Flesh.

Destroying a Demonic Altar will "bless" your world with decreasing amounts of ore, starting with Cobalt, then Mythril, then Adantium and then starting the loop again. This means that it will convert random blocks into whatever ore it is blessing your world with.

It will also place Ebonstone/Pearlstone in random locations underground, allowing Corruption/Hallow to spread faster and from multiple locations.

Source: Terraria Wiki

You're looking at a Demon Altar. These act as crafting stations for pre-Hardmode boss summoning items, and the Night's Edge, the most powerful pre-Hardmode sword. They spawn randomly underground when the world was generated and are commonly found in the caverns under corrupted areas.

They cannot be moved (if you try to hit them with a hammer you get damaged with half your health as you discovered), but after you reach Hardmode they can be destroyed using the Pwnhammer to bless the world with Hardmode ores, an important step in progressing further in game.

That is a Demon Altar, it has various uses. One is that it is a crafting station for certain rare items. The boss summoning items and one strongest pre-hardmode sword.

They are one of the few crafting stations that are not movable. And most of the game they are not even destroyable. Attacking them causes the attacker to lose life.

When the wall of flesh has been defeated, the demon altars can be destroyed using the pwnhammer or the hamdrax.

Each Altar that is destroyed will add new sources of hardmode ores to the game. The first destroyed will add Cobalt, the second Mythril, and the third Adamantite. The fourth adds Cobalt again, but less than the first altar destroyed. Read more about Hardmode, here.

As the Altars are a important crafting station, and they cannot be moved. It can be smart to build your main base near one of the altars. And not destroy this altar, as you want to keep your crafting station.

You can find altars below ground, and they are more common near corruption areas. You can usually find a few near shadow orbs. You can easily find them because they emit a small amount of light.

Those are not monsters, they are Demon Altars... and awesome find on 3 of em at once :)

I also would like to add to other answers, that the only way you could destroy/mine/pickup a Demon Altar, is if you have the luck and found the altar on top of a sand block. Destroying the sand block destroys the altar, as it falls down. According to the wiki page it's considered as a bug, but I hope they won't fix it, as it looks good in my house :)

Also, the (currently) strongest weapon, Night's Edge is crafted on a Demon Altar.