Here is an example.  In a book that describes, say, Pascal, there
will be productions that look like this:

<statement> -> <variable> := <expression>

<expression> -> <variable>

<variable> -> <identifier>

<identifier> -> <letter> | <letter><alphanumstring>

<letter> -> a|b|c|d|e|f|g|h|i|j|k|l|m|n|o|p|q|r|s|t|u|v|w|x|y|z

<digit> -> 0|1|2|3|4|5|6|7|8|9

<alphanumstring> -> <alphanum> | <alphanum><alphanumstring>

<alphanum> -> <letter> | <digit>


(These would only be a few of the hundreds, or perhaps thousands, of
productions which define the CFG for Pascal.)

However, the grammar would generate lots of "incorrect" programs.
Using just the above productions, we could generate the string

"height0 := maxheight" from the grammar symbol <statement>.

If Pascal were truly context-free, then "height0 := maxheight" could
be freely placed in any position of the program were a statement is
allowed, for example, we could write:

 program incorrect(input,output);
  begin
   height0 := maxheight
  end.

But you know that's wrong.  Why?  Because that string is only a valid
statement if the variables height0 and maxheight have been declared,
and the statement is in the scopes of both of those declarations, and
if their types are consistent.

For example, the following is a Pascal program:

 program correct(input,output);
  var
   height0,maxheight:real;
  begin
   height0 := maxheight
  end.

It turns out that the requirement that identifiers be declared makes
giving a context-free grammar for Pascal impossible.  The so-called
"context-free grammar for Pascal" thus generates all Pascal programs,
but also generates many incorrect "wannabe" programs that are in error
because of violation of the declaration rules.  We call this the
"underlying CFG for Pascal."

There is, of course, a "true" grammar for Pascal, but it's not a
context-free grammar.  For more on this subject, there is a book
by Pagan that goes very deeply into advanced grammars.  In our course,
we will deal with only one other class of grammars, which we call
"general grammars," or "unrestricted grammars," or "phase-structure
grammars."  (These are all different words for the same thing.)
This is the most powerful class of grammars, meaning that if any language
can be generated by any grammar whatsoever, it can be generated by one
of those.