Using the Bitcoin digital currency, do I have to be a part of their cluster?

I want to be able to have a digital currency for my game community. It would be its own currency and people could buy games, in-game items, or physical items with it. They could sell these items to each other after use through our market.

So what I dont understand about Bitcoin and many other open source digital currencies is, can I do this on my own, or does the open source software make you be part of the bitcoin economy?

You can always start your own chain, you can tell this by how many coins there are right now. Litecoins, bitcoins, terracoins, namecoins, the list goes on. So, yes, you can have your own version for your own purposes, it doesn't effect other chains.