Before you start mining Bitcoin, it's useful to understand what mining really means. Mining is the process of running SHA256 double round hash verification processes in order to validate transactions and provide the requisite security for the public ledger of the bitcoin network. The speed at which you mine is measured in hashes per second.
The bitcoin network compensates miners for their effort by releasing bitcoin to those who contribute the needed computational power. This comes in the form of both newly issued coin and from the transaction fees included in the transactions you validate when mining. The more computing power you contribute, the greater your share of the reward.
To begin mining, you'll need to aquire bitcoin mining hardware. In the early days of bitcoin, it was possible to mine with your computer CPU or high speed video processor card. Today that's no longer possible. Devices based on custom ASIC chips who's performance offers up to 100x the capability of older systems have come to dominate the industry. Mining with anything less will consume more in electricity than you're likely to earn. It's essential to mine with purpose built bitcoin mining hardware. Several companies such as Butterfly Labs or Avalon offer excellent systems built specifically for bitcoin mining.
The next step to mining is to set up a Bitcoin wallet or use your existing Bitcoin wallet to receive the Bitcoins you mine. A Bitcoin wallet is like a traditional wallet and can be software, mobile, web-based. Bitcoin hardware wallets are also available. Bitcoins are sent to your wallet by using a unique address that only belongs to you. The most important step in setting up your wallet is securing it from potential threats by enabling two-factor authentication or keeping it on an offline computer that doesn't have access to the Internet. Wallets can be obtained by downloading a software client to your computer.
For help in choosing a wallet, go here.
Once you've received your bitcoin mining hardware, you'll need to download a special program used for mining. There are many programs out there that can be used for Bitcoin mining, but the two most popular are CGminer and BFGminer which are command line programs. If you prefer the ease of use that comes with a GUI, you might want to try EasyMiner which is a click and go windows/Linux/Android program.
For more detailed information on bitcoin mining software, click here.
Once you're ready to mine, we recommend joining a mining pool like eclipsemc or eligius. Mining pools are groups of miners working together to solve a block and share in it's rewards. Without a mining pool, you might mine for over a year and never earn any bitcoins. It's far more convenient to share the work and split the reward with a much larger group of miners.
For a full list comparing all mining pools, go here.