I am looking for an inexpensive place to stay in New Orleans for a few
days. Something like a Catholic monastery, YMCA, on a boat, etc.

Here is one, $14.00/night in a dorm style room, $30/night for a 2
person room. If you have a suitcase, rather than a backpack, they
won't care.
"India House Backpackers hostel"
http://www.indiahousehostel.com/

Here is a website that lists a number of different places, most are
$60 and up for 1 person 1 night.
"Discount Hotels New Orleans"
http://www.aanhotels.com/bin/webcgi.exe?agentid=273&citypref=NO

Here is another website similar to the one above:
"New Orleans Cheap hotels"
http://www.allcheaphotels.com/new_orleans/

Here is a list of other hostels in New Orleans
"Hostels in New Orleans"
http://www.gumbopages.com/no-hostels.html

p.s. I have stayed in a number of hostels, and I have (almost) always
met a lot of really cool people, and had a really great time, for next
to nothing. But if the hostel sucks, then in NOLA you can easily stay
out all night and party, and only go back to the hostel to check out,
and get your stuff. :)
Have fun.

Thanks so much for your prompt and informative answer. Sorry it took
me so long to respond. I'm just learning the ropes of all this.

Hi there.. you could also check http://www.hostelbooking.com/ :)