A chargeback happens when a billing on a card is disputed. So, if you buy something and pay by card, and then subsequently you tell your card company to claw the payment back, that's a chargeback.
Usually, credit card companies don't ask for proof that the chargeback is justified. They don't even ask for evidence. They just do it.
The new rule is that, as well as costing the vendor the amount that has been clawed back, Barclaycard also impose a fine of £10 (they call it a "processing fee").
Even though it's so easy to do, you shouldn't do a chargeback lightly. If your credit card number becomes known as one that does chargebacks, then merchants are within their rights to refuse to accept payment from you via credit card. Which could be a nuisance - you really don't want to get onto any credit card Blacklist.
I've never done a chargeback. I usually buy things via Amazon or Ebay, using Paypal. Amazon are, I think, quite reputable and I believe that they'll respond appropriately if I ever need to complain to them. The small, far-off companies that I buy from via Ebay - well, there's Ebay's dispute system, which I've used a couple of times (usually when the wrong thing is sent, or if something is totally unsuitable for the purpose it was sold for) and that's always worked for me. The vendors on Ebay don't want to get a poor review. If that were to fail, Paypal would spring into action to deal with the problem, and get me a refund. HobbyKing, when they sent the wrong stuff to me, sorted it out ... eventually. So I've never needed to do a chargeback.
But this £10 fine on card chargebacks is going to have a bit of a chilling effect on the ability to buy small items over the internet. Look at it from the vendor's point of view. He's selling an Acme Widget for £0.99, and people pay by various means, mostly Paypal and credit cards. If the buyer stiffs him by subsequently doing a chargeback on the transaction, under the old rules, the vendor lost £0.99. Under the new rules, the vendor loses £10.99.
Credit cards suddenly got a bit less useful.