You got quite a few things wrong there RabidChimp.


First off, char[9] holds only 8 characters not 10.
Second, you're presenting <string.h> as if it were an option.
Third, it's not <windows.h>, it's <cstdlib>.
Fourth, you have the wrong explanation for the ignoring of the Y/N.

Ok, so let's see.
Code:
#include <iostream> // The .h standard headers are deprecated and therefore evil.
#include <string> // for the string class
#include <cstdlib> // for system()
using namespace std; // don't worry about that now, you'll learn about it soon enough

int main()
{
	// global variables (outside of any function) are dangerous and a bad habit to get into
	// character arrays are a ready source of errors
	// Use the string class for now.
	string name;
	char accept;
	cout<<"Please enter your name" << endl; // Whenever you're finished with a block of text and
	// doing something else next (like reading input) you should use endl instead of the last \n.
	// You'll learn about that too if your course or book is any good.
	cin>>name;
	cout<<"You entered "<<name<<"\n";
	cout<<"Was this Correct? (Y/N) ";
	cin>>accept;
}
As for the Y/N thing: it's a follow-up of the single-character error.
Visualizing what happens in your program should help.
cout<<"Please enter your name \n";
This requests cout to write that string to the screen. But cout doesn't do that. In order to be more efficient, cout collects a bunch of characters in a buffer and then outputs them all at once. The entire string fits into the buffer so cout doesn't write it to the screen just yet.
You can force cout to write the buffer to the screen anyway by flushing it. That's what endl does btw: it outputs a newline and then flushes the buffer.
cin>>name;
cin is asked for a character. cin looks in its own (input) buffer and sees that there are no characters. It therefore enters wait state, which also means that it flushes the associated output buffer: the string sent to cout is only now written to the screen. You enter "Anjari" and hit enter. The enter causes cin to wake up and check the buffer again. Voilá, there is an 'A' waiting. cin returns the A.
Next, cout is again requested to output something. Again it all disappears in the buffer.
Then cin is asked for a second character. But this time it doesn't have to wait: there is still the 'n' from "Anjari" waiting.
So the output buffer doesn't get flushed and the string never appears. Neither does cin wait for input. The program ends and you're left in the dark.

I hope this was a good explanation, if not feel free to say so.