An "altcoin" is any digital currency created as an alternative to Bitcoin. Many were created as a way to try out new ideas or to improve on aspects of Bitcoin that were seen as limitations. Since Bitcoin is an open source project, the source code can be viewed in its entirety, and if someone wants to they can save a copy of it and make changes and additions, then introduce it as an entirely separate currency. This is how the altcoins came to be.
Most altcoins can be put into one of two general categories, depending on how new coins are generated. Many altcoins are "mined" using the same Proof of Work method as Bitcoin. But some others use an alternate method called "Proof of Stake", which is arguably the Altcoin movement's greatest contribution to the cryptocurrency world. In this method, new coins are given randomly to those who have full-node wallets on the network that are holding coins- the stakeholders in the coin, hence the name. The mining process that generates new Bitcoins is very energy-intensive and requires specialized hardware, and many have expressed concerns that this would lead to ever-greater centralization of mining power and control of the network. So Proof of Stake was created as a more inclusive, less centralized alternative. This method has drawn criticism of its own, however- since it rewards people who keep coins in their wallets, and gives a greater reward to the ones with more coins, many have been concerned that it would encourage hoarding, and would make people less likely to want to spend their coins.
Here are some examples of altcoins that have either introduced innovative new features or were significant milestones in the evolution of digital currency: