We'll start with the basics: octopus suckers. These do function based on suction alone (well, OK, suction and friction), but they are by no means mere suction cups! The image below is a scanning electron micrograph of an octopus' sucker:
The black line at bottom-right is a scale bar, representing 1mm. The mechanism of the sucker is actually driven by three distinct muscle groups which must work together to increase the volume inside the cup of the sucker, decreasing the pressure inside and thereby causing outside water pressure to create adhesion. Once more, proof that you don't need to understand pressure differentials to be able to use them (hooray for the blind watchmaker!). Additionally, that rough texture you see on the inside is due to what are called "denticles," a fancy way of saying "tiny bumps," which increase friction and prevent the suction cup from sliding around. The octopus' strenuous, on-the-go lifestyle is very stressful to these denticles, and so the sucker linings must be shed occasionally. Put a bunch of these together on a limb, and you have the octopus' ubiquitous tool: simple in concept, but executed to a very complex degree. Octopi use their suckers for an enormous number of tasks, and they are correspondingly dextrous, even capable of bending at the rim to grip items smaller than the sucker itself!
Forget those microscopic hairs you saw in the movie; forget the pseudoscience of "manipulating inter-molecular attraction;" this is how Spider-Man clings to walls. That spider must have been infused with octopus DNA in addition to being irradiated!
The squid's sucker is a wicked claw next to the octopus' delicate hand. Squid actually have several different kinds of suckers, different ones on the arms than on the tentacles (the eight arms are short, thick, and suckered all over their inner surfaces; the two tentacles are longer, slimmer, and only suckered at the ends). Here is an arm sucker from a North Atlantic squid:
Note that the inner surface is more tooth-like than that of the octopus - those tiny, chitinous teeth are downright gummy when compared to the suckers on the tentacles, though:
These things are downright vicious! Squid use those fang-like circles not only to grip, but to actively latch onto their prey. Note the scale, and how tiny these little guys are: that "µm" symbol you see means we're talking about microns, or thousandths of a millimeter. And here's how densely they're packed:
The largest species of squid are known to fight off the whales that try to eat them (edited; I had it backwards before). Here is a picture of some battle-scarred whale skin. Pretty brutal, huh?
So yeah, I hope you'll never look at calamari the same way again. In fact, I hope you'll feel even cooler for eating such a frighteningly finely-tuned animal. And just in case you're not scared yet, check this out:
The above image is one of the Honorable Mentions from the National Science Foundation's 2007 Science & Engineering Visualization Challenge. You should check out the rest, they're downright groovy!
Sources and additional links:
NSF S&E Visualization Challenge: Seriously, I could do a 101 Interesting Things entry on anything in there!
The Structure and Adhesive Mechanism of Octopus Suckers: This is a bona-fide, Serious Business journal article, and it's pretty accessible (depending upon your vocabulary).
North Atlantic Squid: Sucker Structure and North Pacific Squid: Sucker Structure: The first is where I got the squid micrographs from, the second has similar (and similarly interesting) photos, and both have side-by-side shots for direct size comparison.
Treasures (Museum Victoria): Hooray for online exhibits!