A Simple Explanation of Mining
Mining describes the calculation of new blocks in a Blockchain. Every information, which is going to be saved into the Blockchain, is collected in blocks and then added to the Blockchain. These calculations need enormous computational power and are processed by independant volunteers all over the world.
Technology
All informations, like a transaction, will be written into a list. This list of transactions will be collected into a block. This block also contains more information, like its unique, digital "fingerprint". This fingerprint is calculated from all informations of the block. This fingerprint will change completely if one small information of the block is changed, caused by mathematical functions. Because all blocks are connected with this fingerprint, all follwing blocks will change, if one block changes his fingerprint and the blockchain will be invalid. This creates the high security of blockchains and is the reason why a blockchain is trustworthy.
Distributed Network
Every user of a Cryptocurrency sends his transactions in the network, in which all miners recieve the new transaction. Every Cryptocurrency has its own, independant network of voluntary miners. These miners recieve all transactions or informations, which are going to be calculated into the Blockchain. They then calculate the next block and if they are finished, they send the extended blockchain to ever other user.
Computational power
The need of computational power is extremely high and this is planned like that. Every block contains a provable work, the computational work of a computer. If more miners join the networtk, the difficulty of finding new blocks will be increased. This ensures a high stability and Cryptocurrencies are not likely to loose their value due to a high inflation.
This concept, a high computational workload as a proof for the authenticity of the calculated informations, is called "Proof-of-Work". The proof trough performed work.
The high workload of the process is the reason, why it is practically impossible to manipulate the blockchain. To manipulate the Blockchain, you would need more than half of all the networks computational power combined. Even a company like Google would not be able to achieve that.
Another concept is not relying on the performed work as a safety, but on how big the share of one miner is of a Cryptocurrency. All transactions getting calculated too, but the computational power which is needed is a lot lower. If you can trust a calculated Block is determined by the amount of units one miner has, the higher his stake, the more the network trusts him. This concept is called "Proof-of-Stake".
Reward
Every miner gets rewarded for the work of their computers. If a new block is calculated, it also contains a transaction to the own account. In the Bitcoin network, this reward is 12.5 Bitcoin. Today, this would be worth around $40000. But to calculate a block with a private PC is nearly impossible, because the computational power needed is to high. This is the reason why many big companies decided to build gibt computer farms, which are only designed to calculate new blocks for Cryptocurrencies. They recieve a big part of the total reward which is available every day. The biggest Mining-companies earn several million dollars every month. Recently, "Genesis Mining" rented a whole Boing 747, to ship hardware to another place. The money lost when the hardware would not be mining Ethereum, would be bigger than the cost of the plane. The total reward which is available in the Ethereum network totals around 7.2 million dollar every day.
Are big companies damaging the network?
No, because none of the biggest companies is big enough to reach a critical limit, at which they would get too much control over the network. Big companies are a important factor of the networks of a Cryptocurrency, because the normal user is not using his own computer to support the network. Big companies compensate all the needed computational power.
Mining yourself?
Mining on private computers is not profitably because of low power and efficiency. Energy cost will often exceed the rewards. Another option is to join a mining network, which increases its computational power by connecting a lot of computers around the world. Platforms like Minergate provide simple applications to join a mining network and to try mining yourself.