The cryptocurrency that started it all is still the most popular, despite its problems. At around $41 billion, it has the biggest market cap. It's been on the market for eight years, it's widely used around the world, and no one found an easily exploitable weakness in the way it works yet.
Bitcoin is both a store of value and a payment system, letting users easily send and receive bitcoins (also known as BTCs; digital tokens that are essentially Bitcoin's currency). It also has a somewhat sexy history, due to the unknown identity of its maker, a person or group of people called Satoshi Nakamoto, which disappeared about two years after launching the project in early 2009.
Bitcoin is based on an invention called the blockchain, which is also the basis of the vast majority of cryptocurrencies out there. Understanding the concept is not trivial, but is necessary in order to get a sense of what all these other cryptos are about.
Blockchain is a distributed database that stores all the transactions on the network in chunks of data called blocks. Every user has a copy of the blockchain (it's a bit more complicated than that, but that's the gist of it). So if Peter sends 1 bitcoin to Alice, everyone else on the network has that record. Furthermore, each sequential block on the blockchain has a cryptographic, time-stamped trail to the last record, which cannot be forged without everyone else noticing. The network itself is powered by miners, which employ computing power to calculate or "mine" the next block. This has a dual purpose: It generates new coins, and it provides the computational power for the system to record new transactions.
In practice, the blockchain solves several very important problems. It's a fully transparent system of financial transactions, in which every single transaction ever made can be tracked by anyone. It's decentralized, meaning you don't need a central institution, like a bank, to confirm anything. You also don't need bank counters or the people to work them. You don't need vaults to store money. You don't need online banking systems that depend on a different team of developers for every single bank. Bitcoin has it all: It's a way to store digital money, a way to send or receive it, and a way to securely store it for every single user out there. The transactions are fairly quick, secure, and the currency is doing all of it on its own, no maintenance required.
But not everything is right with Bitcoin, though. The block sizes in the blockchain are small and can accept a relatively low number of transactions, so with adoption growth the network has become slower. Fees have also risen. They're optional, but the big backlog of transactions means that if you don't pay the fee, your transaction will take a long time to process. The raging debate on how to fix this remains unsolved, and many feel Bitcoin is paralyzed because of it. Adoption of Bitcoin, while constantly growing, is far from ubiquitous; you still can't go to Amazon or eBay and buy stuff with bitcoins (at least not directly). And other cryptocurrencies have solved some of these issues and expanded to allow for far more advanced utilization.
