I was looking up the difference between the two classes and this point came up in a lot of the answers with this blog being the source: http://javarevisited.blogspot.com/2010/1
The best way is to probably look at the source for each class as implemented by the Open JDK implementation for each class; that way, you can get your answer straight from the horse's mouth, as it were :-)
That aside, essentially, "fail-fast" in this sense means that an Iterator over a HashMap will throw an exception if it detects that another thread has modified the targeted HashMap - if you look in the source for HashMap, you will see this is done by simply checking a counter for the number of expected modifications. If the modification count is different than the Iterator expected, that means that someone else has come in since the last check and messed around with the HashMap, and so the Iterator throws a ConcurrentModificationException.
A "non fail-fast" Iterator wouldn't bother to check, and happily go along it's business in the underlying data structure. Therefore, you gain some flexibility (probably dubious flexibility in this case) in exchange for possibly running into errors later; i.e. attempting to access a value that is no longer present.
As with all fail-fast strategies, the idea is that the earlier an error is detected, the easier it is to recover from or debug.