What is the english word that has all five vowels in it, occuring in the same 
order? I mean, "a" should occur before "e", "e" should occur before "i", etc.  
I used to know this, but now I cannot remember.

The shortest and most common word I found that meets your criteria 
is "facetious".  "facetiously" also contains the vowels in the right order and 
even has a 'y' at the end as an added bonus.  Here is a list of other words 
that meet your criteria in the strictest sense, with no vowels occurring 
outside of their allotted place:

abstemious
abstemiously
abstentious
adventitious
adventitiously
arenicolous
arsenious
cavernicolous

Here are some other words that have all five vowels in the correct order, but 
also contain some duplicate vowels which are out of order:

abstemiousness
abstemiousnesses
adenocarcinomatous
amentiferous
anemophilous
antireligious
argentiferous
arteriovenous
autoecious
autoeciously
facetiousness
facetiousnesses
garnetiferous
sacrilegious
sacrilegiously
sacrilegiousness
sacrilegiousnesses
ultraserious

I found these by matching words in a word list against a regular expression.

here is a longer list of words with the desired property, found in the same way
as the above list, only searching through larger word list, so some of the words
are more obscure.  the word list in question in fact has some phrases too.  

anyway, using the strict constraint -- exactly one occurrence of each vowel, and
with the vowels in order -- i get the following:

abstemious
abstemiously
abstentious
acheilous
acheirous
acleistous
affectious
aleikoum
anemious
annelidous
arsenious
arteriosus
arterious
bacterious
battle it out
brazen it out
caesious
facetious
facetiously
fracedinous
gathering clouds
gathering ground
half-serious
half-seriously
have it out
majestious
parecious
pareciously
placentious
trade discount
trade discounts
tragedious
wandering soul
watering trough
wrapped in clouds
wrapped in thought


with the relaxed constraint -- allowing extraneous occurrences of the vowels --
i get a lot more, many of them long phrases...  for example, the phrase "a woman
without a man is like a fish without a bicycle" fits the (relaxed) constraint,
but that's not terribly interesting.

A great tool for solving crossword puzzles, anagrams, and questions like this 
is an alveary. There are many freeware and shareware ones available on the web.

://www.google.com/search?q=alveary

Bryson's TEA (The Electronic Alveary) is excellent. It used to be freeware but 
is now shareware. 

http://www.bryson.demon.co.uk/tea.html

The old free DOS version can still be downloaded from

http://www.und.nodak.edu/org/crypto/crypto/anagram/tea/

From _Eunoia_, by Christian Bok:

    'Eunoia' is the shortest work in English to contain all
    five vowels, and the word quite literally means 'beuti-
    ful thinking'. _Eunoia_ is a univocal libogram, in which
    each chapter restricts itself to the use of a single vowel.

The whole book appears to be available online,
though I recommend ordering the hardcopy,
a slim yet handsome volume to add to your library.

See http://www.chbooks.com/online/eunoia/new_ennui.html

Sorry, a typo in my previous posting:
"libogram" should be "lipogram".