Synchronous is defined as happening at the same time.
Asynchronous is defined as not happening at the same time.
This is what causes the first confusion. Synchronous is actually what is known as parallel. While asynchronous is sequential, do this, then do that.
Now the whole problem is about modeling an asynchronous behaviour, because you've got some operation that needs the response of another before it can begin. Thus it's a coordination problem, how will you know that you can now start that operation?
The simplest solution is known as blocking.
Blocking is when you simply choose to wait for the other thing to be done and return you a response before moving on to the operation that needed it.
So if you need to put butter on toast, and thus you first need to toast the bred. The way you'd coordinate them is that you'd first toast the bred, then stare endlessly at the toaster until it pops the toast, and then you'd proceed to put butter on them.
It's the simplest solution, and works very well. There's no real reason not to use it, unless you happen to also have other things you need to be doing which don't require coordination with the operations. For example, doing some dishes. Why wait idle staring at the toaster constantly for the toast to pop, when you know it'll take a bit of time, and you could wash a whole dish while it finishes?
That's where two other solutions known respectively as non-blocking and asynchronous come into play.
Non-blocking is when you choose to do other unrelated things while you wait for the operation to be done. Checking back on the availability of the response as you see fit.
So instead of looking at the toaster for it to pop. You go and wash a whole dish. And then you peek at the toaster to see if the toasts have popped. If they havn't, you go wash another dish, checking back at the toaster between each dish. When you see the toasts have popped, you stop washing the dishes, and instead you take the toast and move on to putting butter on them.
Having to constantly check on the toasts can be annoying though, imagine the toaster is in another room. In between dishes you waste your time going to that other room to check on the toast.
Here comes asynchronous.
Asynchronous is when you choose to do other unrelated things while you wait for the operation to be done. Instead of checking on it though, you delegate the work of checking to something else, could be the operation itself or a watcher, and you have that thing notify and possibly interupt you when the response is availaible so you can proceed to the other operation that needed it.
Its a weird terminology. Doesn't make a whole lot of sense, since all these solutions are ways to create asynchronous coordination of dependent tasks. That's why I prefer to call it evented.
So for this one, you decide to upgrade your toaster so it beeps when the toasts are done. You happen to be constantly listening, even while you are doing dishes. On hearing the beep, you queue up in your memory that as soon as you are done washing your current dish, you'll stop and go put the butter on the toast. Or you could choose to interupt the washing of the current dish, and deal with the toast right away.
If you have trouble hearing the beep, you can have your partner watch the toaster for you, and come tell you when the toast is ready. Your partner can itself choose any of the above three strategies to coordinate its task of watching the toaster and telling you when they are ready.
On a final note, it's good to understand that while non-blocking and async (or what I prefer to call evented) do allow you to do other things while you wait, you don't have too. You can choose to constantly loop on checking the status of a non-blocking call, doing nothing else. That's often worse than blocking though (like looking at the toaster, then away, then back at it until its done), so a lot of non-blocking APIs allow you to transition into a blocking mode from it. For evented, you can just wait idle until you are notified. The downside in that case is that adding the notification was complex and potentially costly to begin with. You had to buy a new toaster with beep functionality, or convince your partner to watch it for you.
And one more thing, you need to realize the trade offs all three provide. One is not obviously better than the others. Think of my example. If your toaster is so fast, you won't have time to wash a dish, not even begin washing it, that's how fast your toaster is. Getting started on something else in that case is just a waste of time and effort. Blocking will do. Similarly, if washing a dish will take 10 times longer then the toasting. You have to ask yourself what's more important to get done? The toast might get cold and hard by that time, not worth it, blocking will also do. Or you should pick faster things to do while you wait. There's more obviously, but my answer is already pretty long, my point is you need to think about all that, and the complexities of implementing each to decide if its worth it, and if it'll actually improve your throughput or performance.
Edit:
Even though this is already long, I also want it to be complete, so I'll add two more points.
In our example it would be like starting the toaster, then the dishwasher, then the microwave, etc. And then waiting on any of them. Where you'd check the toaster to see if it's done, if not, you'd check the dishwasher, if not, the microwave, and around again.
I don't really understand how we got there. But when it comes to IO and Computation, synchronous and asynchronous often refer to what is better known as non-overlapped and overlapped. That is, asynchronous means that IO and Computation are overlapped, aka, happening concurrently. While synchronous means they are not, thus happening sequentially. For synchronous non-blocking, that would mean you don't start other IO or Computation, you just busy wait and simulate a blocking call. I wish people stopped misusing syncronous and asynchronous like that. So I'm not encouraging it.