Here are some examples of altcoins that are mined using a "Proof of Work" method similar to Bitcoin:
Namecoin was the first altcoin to be created, and is based on the codebase of Bitcoin- with the same parameters as far as the number of coins, how they are mined, and how often they are generated. What sets Namecoin apart is its ability to store names and data on the blockchain. Its best known function is to register domain names for the .bit domain, creating a decentralized DNS.
Litecoin was created by a former Google employee named Charlie Lee. Like Namecoin, it started as a fork of the Bitcoin codebase- but several significant changes were made, making it the first truly unique altcoin. First, the total number of coins to be created was increased to 84 million- four times as many as Bitcoin. Also, the time between newly generated blocks was decreased to 2.5 minutes, as opposed to 10 for Bitcoin- which allows for faster confirmations of transactions. Most significantly, though, it uses a hashing algorithm known as scrypt, which is different than the one used for Bitcoin (sha-256). For a long time this made Litecoin resistant to the powerful, speciallized ASIC miners that had taken over the Bitcoin network, and made Litecoin very popular among miners who were left in the dust when the Bitcoin mining difficulty skyrocketed.
Started as a joke currency, Dogecoin was inspired by the Doge Internet meme. It was created by a programmer named Billy Markus who hoped to create a fun cryptocurrency that could reach a broader demographic than Bitcoin.
Dogecoin has become widely used online as a "tipping" currency, being given freely to other users on forums and social media- even for acts as small as making a good comment on a site like Reddit. This has led Dogecoin to be known as the "antithesis" of Bitcoin, which is known for being hoarded by people who obsess over the price. Aside from giving to each other, the Dogecoin community has also made headlines with a few Dogecoin-based fundraising efforts. They raised money to help the Jamaican Bobsled Team get to the 2014 Winter Olympics, and in a fundraiser called "Doge4Water", they helped build a well in Kenya. They also raised money to sponsor a NASCAR Driver named Josh Wise.
At its foundation, Dogecoin is based on the codebase of another digital currency called LuckyCoin, which in turn is based on Litecoin. One significant difference though, is there isn't a limit to how many Dogecoins can be produced, which makes it inflationary like most "fiat" currencies. As of February 2015 98 billion coins are in circulation, with about 5.256 billion being added per year. But, since the number of coins being added is the same every year, their percentage of the total supply will decrease over time.
Formerly known as DarkCoin, the goal of DASH is to add anonymity to digital currency. It does this by using a technique called "DarkSend" to mix the records of coin transactions, making them untraceable. With such a strong focus on privacy, the developers thought the name DarkCoin may make people wonder about what it's being used for and slow down it's adoption- so they re-branded it to DASH, short for "Digital Cash", which they say is a better description of their vision for the coin.