The technical explanation of Mining (as defined by the bitcoin mining wiki) is the process of adding transaction records to Bitcoin's public ledger of past transactions. This ledger of past transactions is called the block chain as it is a chain of blocks. The block chain serves to confirm transactions to the rest of the network as having taken place. Bitcoin nodes use the block chain to distinguish legitimate Bitcoin transactions from attempts to re-spend coins that have already been spent elsewhere.
In layman terms, Bitcoin mining simply involves using powerful computers to solve complicated math problems. The computer (or computers, in the case of a Bitcoin mining pool) that solve the problem the fastest are awarded a certain amount of Bitcoins. The video below gives a good overview of how cryptocurrency mining (in general works).
If you want to try your hand at Bitcoin mining, you really need to do your homework. Doing your own mining (as opposed to Cloud mining -- discussed later) is rarely a profitable endeavor and is often done purely for the experience and hobby of it. I set up my own personal mining rig (as described in my blog article here) to get the experience of doing it. I am, by no means, making any real money at it. It's just fun.
There are two primary types of mining hardware. The ASIC miner and the CPU / GPU Miner.
An application-specifc integrated circuit (ASIC) Miner is a device that is specifically designed for calculating hash values. ASIC Miners specialize in the SHA-256 algorithm used in the Bitcoin blockchain algorithm. Because of this, ASIC Miners are typically exclusively used for Bitcoin mining. The hardware can be quite expensive (some units costing upwards of $5,000) and consume significant power (some draw more than 2000 Watts of power). Because of their computational power, they are the most profitable units to run. However, because of the incredibly inexpensive power costs in places like China and Iceland, it still takes a long time to recover your costs and really become a profitable device. Check with any bitcoin mining calculator before making a purchase.
One of the more popular hardware ASIC Miners is the AntMiner series. This picture is of the AntMiner S9 and it costs a little over $3,000.
This unit is capable of performing 13.5TH/s (Terahashes per second). This means that it can do 13.5 * 10^12 hashes per second. This makes this device one of the fastest computational machines available today.
Another popular hashing algorithm (as opposed to SHA-256) is scrypt. Scrypt mining is more efficiently processed through modern Graphics Processing Units (GPUs) or even with modern CPUs. Although CPUs are capable, believe it or not, GPUs are typically better at these calculations because they are more geometric in nature and most modern graphics cards are designed to handle geometric calculations more efficiently than CPUs.
Scrypt mining is primarily used in "Alt Coin Mining" like Litecoin Mining or Ethereum Mining. If you are going to pursue cryptocurrency mining for any form of profit, you should consider Scrypt mining and then using a Bitcoin Wallet to store your Alt Coins and, possibly, convert them into Bitcoin.
So if you are this far in, you are probably determined to try your hand at mining. In order to participate in Bitcoin mining, you need to understand the different types of mining. Depending on your goals you have set for your mining efforts, different types of mining may be more appropriate for you than others.
Because of the growing complexity of the math problems (or blocks) miners are being asked to solve, many computers share the workload and try to solve the blocks together. These computers, working together, participate in what is known as pool mining. Computers that mine as part of a pool have a greater chance of solving the problem first (because of the increased computational power). The down side is that when the block is solved, each computer that participated in the pool receives a portion of the mined Bitcoins that is proportional to the amount of computations they did to solve the block.
Let's look at an example. Two computers in a pool, working together. Computer 1 has twice the computing power as computer 2. Therefore, when the block is solved, computer 1 will get 66.7% of the profits and computer 2 will get 33.3% of the profifts.
This is a good solution for people who want to own their mining hardware and want to have a slow and steady payout of coins for the work they are doing.
If you read the explanation of pool mining, you probably have a pretty good idea of what solo mining is. Solo mining is when a single computer tries to solve a block on it's own and, if it's successful, gets all the profit for itself.
Solo mining is akin to playing the lottery and winning. When you are competing against super computers and mega mining pools, the chances of you solving a block before they do are very very small. On the other hand, if you manage to pull it off, you would receive the entire amount of coin given for solving a block.
In the case of Bitcoin, 10 Bitcoin (at the time of this writing) would net the miner about $42,000 in profit.
Cloud mining is good for people who want to participate in Bitcoin mining (or another coin, for that matter) but don't have the hardware to do the mining or don't want to spend the time and energy to buy, build, maintain and power the hardware.
Cloud mining involves signing up for a cloud mining service and purchasing computational power (called hash power). This hash power is typically purchased under a time limited contract but will run 24x7 for you during that contracted period.
As with pool mining, the more hash power you purchase, the bigger a piece of the reward you will receive as a result of solving a particular block.
Bitcoin (or altcoin) mining can be a fun hobby or a serious way to generate revenue. It really boils down to how much time, energy and money you want to invest. Technical expertise is required should you want to do mining with your own hardware so make sure you spend some time doing you research.
Recently, I set up a Scrypt mining rig in my house to learn how the process works. You can see the results of my efforts here.