Every coin address contains both a public and a private key. The public key allows others to send coins to your address, and verifies the signature of the transaction to ensure everything is in order and finalizes the transaction. The private key, on the other hand, allows you to ‘unlock’ and spend your coins. It does this by signing transactions, which tells the cryptocurrency network that you are indeed the owner of the address in which the bitcoins are held and that the transaction is valid. Whoever holds the private key for a coin address is able to spend the bitcoins which that address holds, so in a very fitting analogy, your private key is essentially the key to the safe which is holding your coins. You can also use the private key of an address to sign a message, verifying that you are the owner of the coins held at any given address. This is all secured through mathematics, using asymmetric cryptography.

They are legal in most jurisdictions in the world but there are a small number nation states that have banned its use, such as Ecuador. Wikipedia has a great guide on how cryptocurrencies are treated in all the countries around the world and explains regulatory policies surrounding it. Regulations vary from one border to the next so you should always research your location’s laws before participating in the network.