Halp! I sent a transaction between an ETH account and an ETC account by accident! How can I recover the ether?
Fortunately for me, an account in Ethereum is an account in Ethereum. This means that anyone that has the private key for an account on the ETH chain also has the private key for the account on the ETC chain and vice versa (it's the same key). See I sent My ETC to my ETH address for details. For the rest of this answer, I'll call the type of ether that was sent "etherefrum" and the type that the was expected to be sent the "expecteum". So if you sent ether on the ETH blockchain instead of the ETC blockchain, ETH is the etherefrum and ETC is the expecteum.
If you have the keys for the receiving account (i.e., you sent ether to yourself)
Load up your expecteum wallet and export the private keys. Load up your etherefrum wallet and import the private keys. Gratz! You have access to those etherefrum again.
If someone else has the keys for the receiving account (i.e., you sent ether to someone else)
Have the other person load up their expecteum wallet and export their private keys. Then have them load up their etherefrum wallet to import those keys. Have the other person refund you your etherefrum (if that's what you want). If the recipient is a company such as an exchange or a vendor, you should probably contact their help desk/technical support. Hopefully they're responsive. There's nothing anyone can really do without the private keys associated with the account your etherefrum was sent to. Asking on StackExchange won't get you any results unless someone who works at that company happens to also be on StackExchange (but they should probably be managing their help desk...)