Video - Bitcoin Q and A What happens to our bitcoins during a hard fork?
What do we do with our bitcoins during a hard fork? Where do we store them? Are they safe? A hard fork (planned, accidental, or as a network-based attack) is "a permanent divergence from the previous version of the blockchain, and nodes running previous versions will no longer be accepted by the newest version." Hard forks in Bitcoin have so far not happened often or on a planned basis, because they have monetary consequences.
If a hard fork does happen, your keys will control value (i.e. coins) on both chains because they have the same ancestry. It does mean you have "twice as many coins," but not in the way you think -- more than likely, they will be worth half as much or less. Ponzi schemes and inflation. Bitcoin is designed to be a resilient, self-healing, anti-fragile system, especially when it is being attacked all the time.
If you directly control the private keys to your bitcoins, you're fine --- your coins aren't being invalidated or going anywhere. When the hard fork happens, you can just decide which chain you want to continue with. Wait until things clarify.
If you don't control the private keys to your bitcoins (ex. on an exchange), move them to address that you control. If you don't, whoever controls your bitcoins will be deciding for you, and not all exchanges/ wallets will be supporting both sides of the fork.