Over at Medium.com, Ben Yu has written a cryptocurrency primer called Cryptocurrency 101. It is a long read, but it has much good history and other background information to enhance your understanding of the reason for and value of cryptocurrencies.
Bitcoin was designed, essentially, as a better ‘digital gold’. It incorporates all of the best elements of gold — its inherent scarcity and decentralized nature — and then solves all the shortcomings of gold, in allowing it to be globally transactable in precise denominations extremely quickly.
How does it do this? In short, by emulating gold’s production digitally. Gold is physically mined out of the ground. Bitcoin is also ‘mined’, but digitally. The production of bitcoin is controlled by code that dictates you must find a specific answer to a given problem in order to unlock new bitcoins.
In technical terms, bitcoin utilizes the same proof-of-work system that Hashcash devised in 1997. This system dictates that one must find an input that when hashed, creates an output with a specific number of preceding zeros, among a few other specific requirements.
This is where the ‘crypto’, incidentally, in cryptocurrency comes from. Cryptographic hash functions are fundamentally necessary for the functioning of bitcoin and other cryptocurrencies, as they are one-way functions. One-way functions work such that it is easy to calculate an output given an input, but near impossible to calculate the original input given the output. Hence, cryptographic one-way hash functions enable bitcoin’s proof of work system, as it ensures that it is nigh-impossible for someone to just see the output required to unlock new bitcoins, and calculate in reverse the input that created that output.