Say Alice wants to send money to Bob. She types his bank account number and bank code into her bank's online banking website, sends the funds. It's free. She has a regular account, so she pays somewhere around $5 per month for that and nothing per transaction. She could also opt for no monthly fee and $.25 per transaction, but since rent, utilities, insurance, groceries, etc. all goes through that account, she's better off with the former. So the transaction is free and it is non-reversible, once sent it can only be refunded by the payee. Sufficient funds are checked before the transfer is initiated by the bank.
Now when is this going to be a reality? Wow, this has been actually possible for the last 28 years! In a far away country called Germany. And even if you didn't own one of those fancy BTX terminals back in 1983, you could send money on a slip of paper, also free of charge or with a small fee. And what's a check (or cheque)? I've only once got one, in the late 1990s. It felt ancient.
Now suppose Mallory manages to get her hands on Alice's online banking credentials (either by phishing or by actively installing malicious software on Alice's computer) and sends Alice's money to a little company called Mt Gox without authorisation, then exchanges it for easily-laundered Bitcoins. This isn't an entirely hypothetical scenario; it's one of the reasons that Mt Gox's EU bank account was cancelled and they've been having problems accepting SEPA transfers.
Edit: If German bank transactions really are non-reversible that partly explains why there was such pressure to kick Mt Gox out, though their bank was in France and there are some interesting interactions with SEPA rules in that case. I suspect there's something important we're missing though.