Once upon a time, nobody trusted anybody, contracts were not well enforced and there weren't any banks. Small transactions occurred using small coins (or the exchange of items, with the difference calculated and reconciled in small coins). Large transactions involved gold and silver coins.
Afghanistan and similar places today illustrate well the problem with this approach: gold can be stolen and big floods and mudslides can make it so you cannot get at your coins stored in your house or elsewhere. So in this faraway time and place, it became popular among a certain set in London to leave some gold coins with the goldsmiths, who had guards and strong boxes to store it in, thus reducing the risk of theft while in the city. From there, it was not a large innovation to write a letter to one's goldsmith, to be carried by hand to him, instructing the goldsmith to pay a certain amount of that gold to the person you owed it to.
The next step was not such a large innovation, either. Now, Mr. Darcy could send a letter to his goldsmith telling him to pay a certain amount of money to Mr. Bingley's account with a different goldsmith, and here is where checking is conceived.
It was So Much Easier to write these letters, rather than carrying around gold to pay people when you lost to them at poker or wanted to buy a horse from them or whatever, that it got to be the case that the goldsmiths, oh, let's just call them banks, now, shall we? wound up spending a lot of time at the end of each working day in a room together moving gold back and forth among them, as the letters directed. Of course, they must have quickly noticed that the 50 pounds that transferred from Mr. Darcy's account with Goldsmith Sam to Mr. Bingley's account with Goldsmith Ben was then transferred right back when Frodo Baggins account with Goldsmith Ben paid 45 pounds to Gandalf's account with Goldsmith Sam -- they really didn't need 50 pounds in gold, just 5, to cover the difference, and that was paid by Sam to Ben and the rest were just notations in their respective books. In fact, you could do all this without Sam or Ben actually having 50 pounds in gold anywhere, but we'll get to that on another occasion.
Clearing systems are always about the transfers between financial institutions. You don't see this stuff on the retail side at all, altho you used to, if you turned over the paper checks that you got back from your bank and looked at the stamps. Up the checks went to the Federal Reserve Bank that your bank had an account with, and back it came down. Less visibly, the money moved from one account in the Reserve Banking System to another, where it then wended its way down to the person who deposited the check.
Faster Payments Service in the UK is a relatively new, quite striking innovation in clearing systems. It is not batch oriented -- none of this Ben and Sam get together at the end of the day and figure it all out. It is real time. Unlike a lot of clearing systems devised in the centuries since Ben and Sam met up at the end of the business day, Faster Payments Service is irrevocable. There is no undoing a transaction, which is sort of weird.
If you write a check on your account, and someone else deposits that check, there are a bunch of complicated rules about precisely when the money leaves your account and when it arrives in the other account. Worse, the representation of your account balance obeys a slightly different set of rules than that (that is, if you deposit money, it often shows as being in your account before it has "cleared", which means if something goes wrong and the check doesn't clear, it will "vanish" from your account). This is all non-intuitive and can cause a lot of distress, and a lot of shady characters will engage in nonsense involving the float. Faster Payments is an effort to make it so that the money only leaves your account as it enters the other account -- and it only enters the other account as it leaves yours. If one side can't happen at that moment in time, then the whole transaction waits until it can go all the way through. It is not clearing, really; it is true transactions across financial institution boundaries, accomplished, presumably, by some pretty complex computer linkages which in turn benefit by the comparatively small number of participants (lots of retail participants; few full member institutions, and other institutions operate as third parties through the full members).
It's hard to know when or if we could institute something similar here in the United States. Virtually all of our mobile app innovations are built precariously atop a banking system still mired in multi-day clearing timelines. And Faster Payments is still incredibly new; it's not clear how it will perform over time. It can take a decade or more for the really problematic holes in any given system to be found and exploited ruthlessly by people who we haven't yet written the laws to identify clearly as criminals. But it sure looks like something we're all interested in having. Someday.