Bitcoin mining is intentionally designed to be resource-intensive and difficult so that the number of blocks found each day by miners remains steady. Individual blocks must contain a proof of work to be considered valid. This proof of work is verified by other Bitcoin nodes each time they receive a block. Bitcoin uses the hashcash proof-of-work function.
As noted in Nakamoto's whitepaper, it is possible to verify bitcoin payments without running a full network node (simplified payment verification, SPV). A user only needs a copy of the block headers of the longest chain, which are available by querying network nodes until it is apparent that the longest chain has been obtained. Then, get the Merkle branch linking the transaction to its block. Linking the transaction to a place in the chain demonstrates that a network node has accepted it, and blocks added after it further establish the confirmation.
Electrum is a software wallet that enables you to set up a strong level of security very quickly. During the simple installation process, you are given a twelve word phrase that will allow you to recover all of your bitcoins in the event that your computer fails. Your wallet is also encrypted by default which helps protect your coins against hackers. Electrum is available for Windows, OSX, and Linux and is our recommended software wallet for beginners. Click here to download the right version for your operating system.