Here's the argument, presented in what I think is a clearer way. We're going to proceed with a proof by contradiction (if you don't know what that is, then I'm afraid the Halting problem may be a little bit beyond you at the moment).

Suppose Hal exists. So Hal is a program, such that Hal(p,x) returns "YES" if p(x) halts, and returns "NO" if p(x) doesn't ever halt, for *any and all programs p* and *any and all values x*.

Now we invent a new program Barry. Barry is implemented as follows:

Barry(p) =

if Hal(p, p) = "YES" then RUN FOREVER

otherwise HALT

In other words, if Hal says "NO", Barry halts, and if Hal says "YES" then Barry just runs in an infinite loop forever. Note that we're giving p itself as input - this is allowed because all programs can be stored as data, and all data can be interpreted as a program.

Now let's see what happens if we do this: Barry(Barry). Well, let's go through the definition, replacing every p with a Barry:

Barry(Barry) =

if Hal(Barry, Barry) = "YES" then RUN FOREVER

otherwise HALT

But remember that Hal(p, x) means "return YES if p(x) halts, else return no". So Hal(Barry, Barry) means "return YES if Barry(Barry) halts, else return NO".

Now there are two possibilities: either Barry(Barry) halts, or it doesn't halt. Let's go through the two possibilities.

* Suppose Barry(Barry) halts. Then Hal(Barry, Barry) would output "YES", which means Barry(Barry) will RUN FOREVER - i.e. it won't halt.

* Suppose Barry(Barry) doesn't halt. Then Hal(Barry, Barry) would output "NO", which means Barry(Barry) will halt immediately - i.e. it halts.

So now we have a conundrum. If we assume Barry(Barry) halts, it turns out it doesn't halt after all. But if we assume it doesn't halt, then it does halt. But ALL programs either halt or don't halt, there's no in-between. That means Barry can't possibly exist as a real program. But look again at the definition of Barry:

Barry(p) =

if Hal(p, p) = "YES" then RUN FOREVER

otherwise HALT

There is absolutely nothing wrong with this program. Everything here is easy and well-defined...except for Hal. We only *assumed* Hal exists. All the other bits ("if", "RUN FOREVER", "HALT") are easy to implement. But Hal is the only thing we assumed existed without proof. It follows then that Hal must be the bit that can't exist - if Hal existed, we could construct Barry, and that's not possible, therefore Hal can't exist. QED.