When bitcoin came out in 2009, it was the first and only cryptocurrency. But soon the bitcoin code base was forked to create other coins by changing some parameters and functionality of the original code. Later also completely new coins where developed from scratch by taking the idea of transactions in a distributed ledger or blockchain from bitcoin for coins with other purposes than only money transfers or just with other characteristics than bitcoin. All these coins, other than the original bitcoin, were given the collective name of alternative coins. In short, altcoins or just alts. Today (February 2, 2019) bitcoin is still the biggest coin in market cap, and there exist over 2100 alternative coins according to coinmarketcap.
The first altcoins where the ones that forked the bitcoin codebase and usually changed some parameters like block time and block size, also the type of PoW algorithm was sometimes changed from SHA-256 PoW to PoW algorithms like Scrypt (Litecoin, 2011).
The first ever bitcoin fork was Namecoin (NMC), it was created for the specific use case of being a decentralized DNS and digital identity system. It was also the first to implement merged mining with bitcoin (simplified, merged mining is using the same mining algorithm of another blockchain and like this it can use the same miners of that chain to create blocks for its chain too, with a proof in the main chain. Merged mining should be explained in more detail in a new topic in the wiki). Although it was the first altcoin, its marketcap is not putting it in the top altcoins today.
Also some coins surged that focused on anonimity. The first one being Bytecoin (BCN) that was build form scratch and its CryptoNote algorithm uses ring signatures and one-time addresses. Later, Monero was forked from BCN and the changed algorithm was called CryptoNight.
A lot of other altcoins were created in the same years but it is impossible to discuss all. (But feel free to add altcoins you think are worth mentionining).
Second generation altcoins
I personally don't like to put projects in categories like first generation, second generation, third generation and so on, because it makes it sound like third generation are in any way better than first and second generation projects, and that is definitely not always the case. You don't always need a Swiss army knife just to cut some bread. And so, Bitcoin, being the first and so of the first generation, is still the most tested, most robust, most secure, mostly valued choice if what you need is just a store of value or a money transfer. Not everything needs blockchain and if you just wants a secure and transparent money transfer, you don't need anything more than bitcoin. And the serious projects of the first generation are still being maintained, developed and also still evolve. Also there is no standard to what is first, what is second and what is third generation. Some vendors and projects just use it for marketing, so be warned when you hear about a sixth generation blockchain, I wouldn't trust it. So for me this categorisation is of little significance. But that completely aside, the categorisation is often used, so for simplicity I will also use them to talk about different types of altcoins.
As Bitcoin is the first and most known of the first generation, Ethereum is usually named as the first and the reference of the second generation cryptocurrencies.
Eventhough not always named like that, Bitcoin actually has smart contract capabilities. Even every basic transaction uses this Bitcoin smart contract language, Script, to lock funds for the receiver. The receiver needs its private key to create the unlocking script. Eventhough more complex transactions can be build like this, the language is not Turing complete, so it is limited. And this limitation is exactly what these second generation blockchains solve by offering Turing complete smart contract languages that can be run by a virtual machine on every node. This way they could build general purpose blockchain and smart contract platforms or like Vitalik Buterin of Ethereum envisioned: A world computer.
Ethereum was the first to start building a general smart contract blockchain. This means that not only money transfers can be more easily programmed, but that also any other program can run or any digitalized assed can live trustless on a decentralized blockchain. This gave rise to new tokens being created on top of Ethereum, the ICO boom and tokenization of real-world assets.
But being such a general platform, it can also be used for games like Cryptokitties. And every transaction of every application on the network and all data and rules of these new assets have to be stored on every node too. This general-blockchain-for-all-applications approach is soon to see limitations in scalability, with the blockchain and state size increasing rapidly and the network being overcharged by just some popular applications.
Third generation altcoins
Third generation altcoins are blockchains that have the general smart contract functionality but also try to resolve the scaling issues.
Some proposed ideas to solve this are sharding, sidechains, ecosystems of application-specific blockchains and also in some way PoS to make mining more efficient.
Personally I believe a lot in sidechains and/or ecosystems of application-specific blockchains that can have a separate set of miners or validators. Like this the charge of the network is more spread out across diferent blockchains that can be adapted more to the specific use case and the whole network shouldn't process or store data that it's not interested in. This also reduces the impact of bugs as one chain may be impacted, while other chains stay correct. All this while being anchored to a robust and secure main chain. I see this as a normal evolution in software development, like the evolution of bulky and difficult to maintain monolithic applications to easy-to-maintain and orchestratable, single-purpose, independently managed microservices.
The cryptocurrency and blockchain that has best investigated sidechains and PoS is Cardano (ADA).
And one of the projects developing the way to build ecosystems of application-specific blockchains is Cosmos.
In general altcoins still follow bitcoin movements, if bitcoin goes up the altcoin market usually follows, if bitcoin goes down, market usually goes down. Delays in this following can be used to increase your bitcoin stack. Also news or events of a specific altcoin can influence the price of the specific altcoin and make it deviate from bitcoin.