What happen if I transfer ether from account a to account b on a private chain with just one node and that node is not mining?
The transaction will simply sit in the txpool until a miner picks it up and includes it in a block.
If there is nobody mining, the blockchain will simply not produce any more blocks and thus not process any transactions.
Additionaly, miners can find and produce blocks without including transactions, if desired.
Your question should not be:
What happens if I transfer ether from account a to account b
What happens if I send a transaction to transfer ether from account a to account b
What happens? Well, the transaction remains in the transaction pool and does not become part of the chain. So the Ethers are not transferred.