A bitcoin is a purely non-fiat electronic-only currency that was created to provide a non-centralized virtual currency that people can then store in virtual e-wallets. They aren't paper money, and can't be held in your hand, but are worth as much as many of the most widely used currencies in the world.
Bitcoins also aren't owned by any single entity or central bank, they are purely mathematical in nature, and are produced by computer algorithms. You could use gold mining as an analogy to mining for bitcoins, there is a finite amount that exists, and at first they're easy to mine, but get more expensive and harder to get as the overall supply increases.
Bitcoins are de-centralized, what this means is that people have on their own machines, or on paid websites what are called "e-wallets". These are where people store their bitcoins. So what happens if you don't back up your bitcoins and your computer dies, or you don't host them on a remote server? Well you could view bitcoins like cash, if they're destroyed or lost it's just like burning money or flushing it down the toilet.
Bitcoins are created by people 'mining' on computer equipment. Although this sounds like something really easy to do it's actually very difficult to accomplish. For example, running bitcoin mining software on your personal computer at home 24/7 would take approximately 1 year to create a single bitcoin. So to make it profitable you would need a decent sized server farm, probably using a renewable source of power such as solar panels. A few of the larger producers of bitcoins have been network administrators running a computer network in a school, or large organization where they have zero start up and maintenance costs.
In theory there can only be 21 million bitcoins created to the specification of the bitcoin in it's current state. So based on these statistics, mining bitcoins is probably not a good industry to get into, basically the only people out there who are going to make money off of mining these are individuals who have been doing it from the beginning.
People around the globe are trading bitcoins all the time. There are markets that allow people to trade them for real world fiat currencies. There are even markets that allow people to speculate on the future value of this virtual currency by trading futures contracts.
You must be wary of bitcoins as an currency however, since it is not a fiat currency, which means that it's not backed up a country. This means that it can have wild speculation prices and valuation. Also, if you somehow loose your bitcoins, it's just like burning money, they're gone forever.
One of the biggest advantages of the bitcoin currency is that it's completely anonymous and is extremely hard to tracks where bitcoins are being traded with. This leads many to believe that they are being used, or will be used for illicit activities such as illegal drug sales and money laundering.