The G00gle search page remembers previous search terms.  If I search
for "Antelope", when I type in the first letter, a menu will appear
showing previous searches beginning with the letter "A". As I continue
entering the word the menu will change to show previous searches
beginning with "An" then "Ant" etc etc.  If any of these previous
searches is what I am looking for, I can click on it as a shortcut.
I remember deleting all these previous searches individually by
holding the mouse over each term so that it was highlighted in blue
and then pressing the "Delete" key.  I went through the letters in
alphabetical order and I also did the numbers, brackets and quotation
marks, so I got rid of everything.

While searching today, I noticed that menus appeared showing previous
searches from over two years ago.  I know this because the searches
referred to a trip abroad at that time.  They started with several
different letters of the alphabet, so I didn't miss out on deleting
one letter by mistake.
Previously, after deleting all the search terms one by one, I checked
to make sure that no menus of previous searches appeared.

How could these search terms have come back?
Many thanks if anyone can explain this.

It is called AutoComplete. Tools > Internet Options > Content >
Autocomplete > Clear Forms

Susk-ga,
Many thanks, that's much easier than clearing each entry individually.
My way of doing it was very tedious.

But I am still wondering how previously deleted search terms could have come back.

Becuse they will stay on your computer untill you permantely delete them.

You can turn off autocomplete altogether by unchecking the boxes that
are above the "Clear Forms" button on the Autocomplete page too. Or
you can ditch Internet Explorer altogether and get Mozilla Firefox
which is better anyway.

http://www.mozilla.org/products/firefox/

You probably need to clean up your cookies by go Tools > Internet
Options > General > Under Temporary File click on Delete Cookies
button