A “wallet” is basically the Bitcoin equivalent of a bank account. It allows you to receive bitcoins, store them, then send them to others. There are two main types of wallets, software and web. A software wallet is one that you install on your own computer or mobile device. You are in complete control over the security of your coins, but such wallets can sometimes be tricky to install and maintain.A web wallet, or hosted wallet, is one that is hosted by a third party. These are often much easier to use, but you have to trust the provider (host) to maintain high levels of security to protect your coins.
Mining is the process of spending computation power to secure Bitcoin transactions against reversal and introducing new Bitcoins to the system. Technically speaking, mining is the calculation of a hash of the block header, which includes among other things a reference to the previous block, a hash of a set of transactions and a nonce (an arbitrary number used just once for authentication purposes).
The receiver of the first bitcoin transaction was cypherpunk Hal Finney, who created the first reusable proof-of-work system (RPOW) in 2004. Finney downloaded the bitcoin software on its release date, and on 12 January 2009 received ten bitcoins from Nakamoto. Other early cypherpunk supporters were creators of bitcoin predecessors: Wei Dai, creator of b-money, and Nick Szabo, creator of bit gold. In 2010, the first known commercial transaction using bitcoin occurred when programmer Laszlo Hanyecz bought two Papa John's pizzas for 10,000 bitcoin.