It's duct tape. But some arguments suggest it could be duck tape. And there's the rub, as the Bard once wrote.
The issue is disagreement over the etymology of the product and which name is correct. (For more information on this, check out the NYT article in the sources below.) In order to try and clarify things, we need to look at a little history. For convenience's sake, I'm going to refer to the product as duct tape throughout this article.
First, let's look at what duct tape actually is. Our friends at Merriam Webster define it as "a wide cloth adhesive tape originally designed for sealing joints in heating or air-conditioning ducts". Which is interesting given that wasn't what the product was originally developed for at all. In fact, the biggest cause for disagreement goes back to when the product was first developed in 1942 by a division of Johnson & Johnson for the United States Army. What the Army needed was something to keep moisture out of ammunition cases. To meet this requirement, a product was developed that applied a rubber-based adhesive to a durable duck cloth backing. The duck cloth, commonly called 'canvas' outside the textile industry, helped provide water resistance so that water tended to bead on it, perhaps similarly to how water rolls off a duck's back. So in 1942 the US Army referred to the product they used as duck tape, because it was made from duck cloth.
References to the product during and immediately after WWII, both inside and outside the military, mostly are to duck tape. Occasionally a more generic name such as 'ladder tape' was used, depending on the seller and the industry they were in. The following example, which fits in the continuity of the SG1 episode 1969, demonstrates this usage.
"Colonel, we're going to need some duct tape in order to fix the bus."
"There's a hardware store down the road, Carter, but make sure you ask for duck tape." He looked at Carter's raised eyebrow. "What? Just because I know it wasn't called duct tape until 1970 doesn't mean I'm old!"
That's right -- the first usage of duct tape only dates to 1970, nearly 30 years after the product was originally created. However, since that time the 't' spelling has become the de facto usage, with the Henkel Consumer Adhesives Company eventually registering the name 'Duck brand duct tape', the current number one product brand in the United States.
So even though the origins of the product seem to point to duck tape possibly being correct, if you go to any store what you're going to find is duct tape. And that's what you should use in anything you're writing that is set after the 1970s. For anything set during WWII or shortly afterwards, you should probably refer to it as duck tape in order to be historically accurate. And you shouldn't use it at all in any story set before 1942.
As an amusing side note, if you're going to use anything to seal HVAC ducts and ducting, the product to use is not duct tape - at least according to a study conducted by the Lawrence Berkeley National Lab, where it was duct tape that failed all the tests conducted around sealing ducts. Of course, that doesn't stop it being used for many other purposes, as our examples demonstrate.
"O'Neill, do you wish me to duct tape Daniel Jackson's mouth?"
"No need, T," Jack said, glancing at a silently fuming Daniel. "I think he gets the message."
It never failed to amaze Jack O'Neill just how many uses Carter could find for duct tape, but how she managed to get it to work with Goa'uld crystals was completely beyond him.
And please don't mix up duct tape with gaffer tape. Gaffer tape is a very specialised product which is designed to be removed without leaving a sticky residue, unlike duct tape which is intentionally very sticky.
In summary, the 't' usage is the correct one -- you won't even find duck tape in the Merriam Webster dictionary. How do you remember? Think about using duct tape to fix your ducts. Only don't do that, because apparently it doesn't actually work. ;-)
Duct Tape at m-w.com http://www.merriam-webster.com/dictionary/duct+tape
Duct Tape at Wikipedia http://en.wikipedia.org/wiki/Duct_tape
NYT on Language - Why A Duck
Duck Brand Duct Tape
Sealing HVAC Ducts: Use Anything But Duct Tape