Malleability is a problem that has been in Bitcoin for many years, it is well known and it is not too hard to make sure a business can't loose money over this. The problem really should get fixed, though, as it is a rather embarrassing issue.
The high level problem is this;
A user can create a transaction, which she signs and sends off to the network in order to get mined. Transactions get automatically assigned a transaction-ID. To uniquely identify that transaction. This assignment is done in a smart way such that the ID is assigned based on the content. This way we avoid having to ask a central server to give unique IDs.
There is a way to change the transaction slightly so it still is a valid transaction that correctly moves money as the original author intended and doesn't break the signatures either. This is called the malleability issue. The small change will have as an effect that the transaction-ID changes, as its based on slightly different content now.
The result is that a company sending a transaction can not be certain that his transaction will have the same ID from the moment is was created to when it finally gets mined in a block.
MtGox claimed that this was a reason they lost funds where users would complain a transfer from the company to them never arrived and the exchange would only check the transaction ID having been mined.
A simple fix is to avoid address reuse and check balances on addresses instead.
There is ongoing research into a framework that uses Bitcoin transactions in a rather interesting way. It allows 2 users or even a group to use low-trust hubs and exchange promises of Bitcoin transactions with each other. This allows a person to make many payments which get compressed into one transaction (settled) on the real Bitcoin chain.
This system would, however, be very vulnerable to the malleability issue and cheating would become much easier for some parties.
How can you malleate a transaction?
Quite technical, Feel free to skip to the solution below.
A Bitcoin transaction of version 1 defines that the transaction-ID is calculated by taking the entire content of the transaction and hashing it.
On the other hand, the thing that protects a transaction from being modified after the fact are signatures that are created for each input. These signatures ignore parts of the transaction. And exactly the difference in what is used for the Transaction ID and what is used for signing an input is where the malleability comes from.
During signing of an input the signatures of all inputs are removed because of the cryptographic truism that signatures can't be part of the thing that is signed.
In adding a signature in Bitcoin 2 things are added. First the amount of bytes that the signature takes, then the actual signature. Both of these pieces of information are removed for signing but used for creating the transaction ID.
The trick here is that we can add some dummy data, or we can encode the length in different ways that mean the same thing. 120 bytes is the same thing as 0120 bytes. But the latter is encoded differently. And thus creates a different transaction ID.
How does Flexible Transactions fix this?
The solution is rather simple and elegant, we removed the imbalance between what is used to sign a transaction and what is used to create a transaction ID.
The way to do this is that FlexTrans simply removes from the body of the transaction the entire set of data that is the input-script (aka signature data).
This is done by physically moving the signatures to the end of the transaction data-structure and defining that the 'body' of the transaction is the one to be signed and that the 'body' is to be used to create the transaction-id.
This same solution also solves the Quadratic Hashing issue.
For more technical information, see the specification.