The first alternative system was not intended to be a currency but instead used the blockchain technology to decentralize domain name registration. Namecoin was introduced in 2011 but the ideas behind it had been discussed one year earlier by Satoshi and others. The system is still in use and anyone is free to register domains under the top domain .bit using it.
Litecoin, which is a more of a straightforward copy of Bitcoin, was also presented in 2011. The mining algorithm had been changed so that GPU:s wouldn't have an advantage over CPU:s like in Bitcoin (read more about the history of mining here). The idea was that anyone should be able to mine the currency using their regular computer. Litecoin also modified the block frequency, from every 10 minutes to every 2,5 minutes. Despite the creator's intentions specific hardware was eventually developed for mining Litecoin so there aren't really any clear advantages over Bitcoin. Anyhow, Litecoin marketed itself as "the silver to Bitcoin's gold" and has managed to stay among the 5-6 most popular cryptocurrencies to this day.
Something that the first copies had in common was that they tried to provide something that Bitcoin couldn't. Peercoin tried to replace Bitcoin's energy heavy proof-of-work algorithm with proof-of-stake and Ethereum was (and still is) a very ambitious project to create a Turing complete programming language for smart contracts.
At some point during 2013 though, an explosion of alternative coins started to happen. DogeCoin, launched in late 2013 was obviously just meant as a joke, based on a popular meme, but had enormous success and still today has a market cap of half a billion dollars.
People began to realize that there was money to be made by launching a new currency and so today there are more than 1000 known altcoins. Most of them completely useless of course.
In other words, this is a djungle. Below are a few of the alternative systems that actually aren't just copies but has contributed with something new to the ecosystem. Follow the links to learn more.