I think you have it a little backwards.
In the U.S., a negotiable instrument (which includes checks, promissory notes, negotiable bills of lading and all sorts of other instruments) is one which may be transferred by endorsement*.
A negotiable instrument has the language “pay to the order of” a named person (or business). That means the bank it is drawn on will pay the named person or anyone the named person “orders” the bank to pay.
How does that named person “order” the bank to pay another person? The named person “endorses” the check on the back. For instance, if the check is payable to Adam, Adam can write on the back “Pay to Bob, [signed] Adam”, and now Adam will likely be the “holder in due course” of the check, who can, if he likes, endorse it over to Carol by signing below Adam’s endorsement “Pay to Carol, [signed] Bob.”
Normally, someone holding a check doesn’t want to go to the bank the check is drawn on, but rather wants to deposit it in his or her bank. However, the check has to be endorsed to “order” the bank it drawn on to pay it to the bank it deposited in. One way this is sometimes done (usually not, but in some corporate endorsements) is to write on the back “Pay to Deposit Bank, [signed]Depositor.” Before electronic clearing, the bank it was deposited would transfer the physical check to a “clearing house” which would sort the checks and sent them to the banks they were drawn on, and along the way there would be all sorts of stamps directing banks and other institutions along the way who to pay the funds to.
Now we get into bearer instruments. These are instruments that are transferred by simply physically transferring the instrument. The ways that this can be done are writing the instrument as “Pay to the order of CASH”, “Pay to the order of BEARER” or “Pay to the order of [name] or BEARER.” With a bearer instrument, no endorsement is needed. You just have to present the actual instrument.
You can convert an instrument into a bearer instrument by “unrestrictively” endorsing it. The easiest way to do this is just signing the back of the instrument with nothing more than your name. It is an “order” to the bank to pay the check to whoever is the bearer.
A slight variation is a “restrictive” endorsement, which is “For deposit only, [signed]Depositor.” With this endorsement, you can deposit it into a bank (or similar depository institution) for credit to the account of the signer, but not transfer it to anyone else other another depository institution for clearing.
It sounds like the Australian system is to have all checks made out as bearer instruments, but with the words “non-negotiable” restricting the transfer to the payee (or, I guess to depository institutions for the account of the payee).