Note: The information provided here on mining is fairly high-level, and is not intended to be completely comprehensive of how mining or blockchains work. It's intended only as an introduction to the basic concepts of mining. You can jump to Mining on Sia if you're already familiar.
Cryptocurrencies are based on a blockchain, which is basically a publicly visible ledger of transactions that can be viewed and verified all the way back to the beginning of the currency and the very first transaction. Transactions are bundled into blocks which are finalized approximately every 10 minutes and include a link to the previous block, hence the name "blockchain" - an ongoing chain of blocks, each referring to the last, through which the entire transaction history can be viewed.
Miners are people running computer systems or hardware that do the math to verify these transactions and "solve" these blocks, also known as mining.
Mining serves two main functions - it advances the network, and it secures the network. If blocks aren't created and solved, no transactions actually occur, so no mined blocks means no network. Transactions are only considered complete when they're finalized and confirmed in a block, and when the network agrees the block is valid. Mining also secures the network by requiring computational power to be used to solve blocks. This means that once a block is solved, it would take much more work to go back and change a previous block (i.e. falsify a previous transaction) because all blocks after it would have to be modified as well, and this would be nearly impossible to do unless one single party controlled the majority of the network's mining equipment (known as a 51% attack).
Why Become a Miner?
Mining usually requires an investment in specialized hardware and electricity costs, so why do people mine? When blocks are solved, the miner that solved the block receives something called a Block Reward, which consists of some amount of the blockchain's currency. For example, the current block reward for mining Siacoins is about 150,000 SC as of this writing. At a Siacoin price of $0.01, that's about $1,500 worth of Siacoin up for grabs about every 10 minutes, or $216,000 a day. On top of the block reward, there are also fees paid by Sia users for each transaction sent, though as far as the Sia network is concerned this amount is usually negligible.
Aside from the financial rewards, miners can contribute to supporting a network by helping to spread their computing power around and ensure that the network remains secure. It doesn't do very much good to earn Siacoins if you can't trust that coins can't be double-spent or your transactions can be changed later because the network is under a majority's control.
The math to verify transactions is fairly trivial, but to make sure blocks occur at approximately regular intervals (i.e. every 10 minutes), a difficulty level exists for each block. This level is flexible and automatically adjusts from block to block - if more miners join the network, more computing power exists and the blocks are likely to be solved more quickly, so the difficulty level is increased. Likewise, if miners leave the network and the difficulty level remains high, blocks (and therefore transactions) would slow down. If enough miners left the network at once, the network could slow significantly and blocks could take hours or days to solve for a while. If all miners left the network, the network would effectively stop.
The difficulty level is usually adjusted by requiring the output of the math done to fit a certain pattern based on the inclusion of a random number in the math. Mining machines try combinations until they find an output that is considered valid and fits the pattern (also called Proof of Work, or PoW). When this is successfully done, the block is "solved", the block reward is paid, and the next block is started.
When you mine, you compete with other miners to solve a block first. This is similar to the lottery - your chances are based on how much computing power you have, the number of other miners and their computing power, and the difficulty of the current block. Mining by yourself (also known as solo mining) means your odds of ever solving a block first are very, very low - just like winning the lottery.
For this reason, mining pools exist. Mining pools allow miners to team up and use their power together to solve a block. A mining pool assigns a miner a piece of work (a share), and the miner completes and returns it. If a block is found by a miner in the pool, the reward is shared amongst all the miners in that pool. Payout schemes are different from pool to pool, but it generally means that all miners make some amount of money all the time instead of only hoping to "hit the jackpot" once or twice. A large enough mining pool will see a consistent regular payout because they have enough computing power to be likely to solve several blocks.
CPU (Central Processing Unit) Mining
CPU mining is generally the slowest and least profitable. A computer's CPU is a very general-purpose computation device, and while it can mine, it isn't optimized to the specific type of math that cryptographic hashes use.
GPU (Graphics Processing Unit) Mining
GPU mining is better than CPU mining because the hardware in a graphics card is more specially suited to do the types of math that cryptographic hashes use. GPUs perform very specific repetitive work, and have more cores available to do so, which is perfect for performing the same type of math repeatedly when trying to solve a block.
FPGA (Field-Programmable Gate Array) Mining
FPGA mining involves a programmable chip that can be configured in a certain way. This configuration can be optimized to run cryptographic hash functions very quickly and with very little power use. However, specialized knowledge is required to properly program and configure these devices. FPGA mining is usually a bridge between GPU mining and ASIC mining.
ASIC (Application-Specific Integrated Circuit) Mining
ASIC miners are the most efficient type of mining devices. They involve using multiples of a physical chip specifically designed to run a given cryptographic hash, so they offer the highest performance in relation to power required. They typically outperform GPU mining by a factor of about 100, so ASICs effectively make other types of mining obsolete once they are released for a certain coin or algorithm.
However, since these devices are so specialized, they can only run one type of hash - for example, a Bitcoin ASIC which hashes SHA256 couldn't mine on Sia, which uses the Blake2b algorithm. They also tend to be expensive (generally thousands of dollars) because designing specialized chips is expensive, and they have no practical purpose once they become obsolete due to a newer, faster version or oversaturation of the network by too many ASICs.
GPU mining was the predominant way to mine Siacoins up until January 2018. Prior to 2018, no ASIC devices existed for the Sia network. Anyone could build a mining computer, or simply put their regular computer with a decent graphics card to work, in order to mine Siacoins. However, as of January 2018, ASICs from a third party have been released for Siacoin and Blake2b. This means that anything less than ASIC mining of Siacoins is now practically obsolete, including GPU mining.
Because ASICs now exist on the network and GPU mining is no longer profitable, no guide is provided for mining Siacoins. Those who purchase ASICs will have access to setup instructions for their particular type of ASIC, which is usually as basic as logging into a web page, selecting a mining pool and entering a wallet address for payouts.
Obelisk: The Sia-Backed ASIC
The Sia development team took pre-orders in the summer of 2017 for their own ASIC device, dubbed the Obelisk SC1. As this topic has resulted in a large history over the last year, we have a complete page of information on Obelisks here.