This means that no one person or entity can turn bitcoin on or off. This is powerful, because it allows bitcoin to exist outside the confines of national borders, anyone controlling interest, or colluding interest.
Bitcoin works on a proof of work (POW) system where different miners solve algorithms to determine who owns what bitcoins. These calculations are stored in a nonce – or block. This allows the network to reach a state of consensus and records that block in a chain of blocks.
There is, in theory, one way to double spend, if one were to buy enough computing power to perform a 51% attack on the network, they could alter the network state accordingly. The computing power needed to do this currently is astronomical, and thus the network is currently secure.
However, there are some issues with the way bitcoin was set free into the wild. Only 3 people were given primary control over bitcoin. Notably, lead developer Mike Hearn recently departed to work on a blockchain project.
The tide has swayed in and out on bitcoin, but as of right now it has lost some favor. Still, the technology which underlies bitcoin is incredible, and it has gone through huge fluctuations in price (from where I first publicly mentioned over 4 years ago – the price of bitcoin was $10 and has gone over $1000, and now hovers around $300-400 price range at the time of writing in early 2016. [update Jan 2017 – the price as doubled and now hovers around all-time high].