If you rank animals by intelligence, you get a nice, orderly progression (leaving aside the question of what intelligence really is, assuming that I know it when I see it). Invertebrates show little intelligence to speak of, with the exception of arthropods, such as bees, cockroaches and spiders, who exhibit remarkably flexible behavior. Still, remarkable though they may be, they are not as smart as amphibians, fish and reptiles, who are, in turn, not as smart as birds. And at the top of the IQ-chain, of course, we find ourselves, the mammals.
But then there is the octopus, the nerdy cousin of the snail. Octopuses  are the only animals I can think of that are fairly intelligent, but are in no way part of the evolutionary axis-of-intelligence formed by the vertebrates. They have a kind of weird, alien intelligence, which is why I think the octopus is the single most fascinating species of animal. (I regret to say that I've never seen one in real life, because all the zoos that I've visited either did not have an octopus or it's aquarium was mysteriously vacant, as cages and aquariums in zoos tend to be.)
Let me give you some numbers, mostly taken from Hochner's delightfully accessible review (2008). An octopus has about 50,000 times as many neurons (brain cells) as a garden snail, which, like the octopus, is a mollusc (molluscs form a rather large phylum, but still). This may not sound terribly impressive, because 50,000 times zero intelligence is still zero intelligence, but you will probably be surprised to learn that an octopus falls somewhere in between a rat and a dog when it comes to number of the neurons. And considerably closer to the dog. You may argue that octopuses are big animals and that it's not absolute, but relative brain size that matters. Perhaps, but also by that measure, octopuses fall well within the vertebrate range.
But even more convincing than these numbers is a study by Fiorito and Scotto (1992), in which they showed that octopuses can learn by observation (but see ). That is, one group of octopuses learned a task, which took them some time to master. A second group of octopuses observed the first group and, sure enough, they were able to perform the task pretty much right away. Just to stress how amazing this is: Try to think of other animals that would be able to do this. Monkeys, to be sure, dolphins, perhaps even dogs and cats, but I don't see a bunny learning by observation.
Still, for all their apparent intelligence, it has proven difficult to show that octopuses have any kind of “eye-hand” coordination, in the sense that they have not been observed to guide their tentacle movements visually in any kind of sophisticated way. This is why a new study by Gutnick and colleagues (2011) in Current Biology is pretty cool. They designed a task that requires a visually guided reaching movement. The octopus reached through a tube. At the end of the tube there were three smaller tubes, one of which contained a food reward. Because the tubes were transparent, the octopus was able to see the food. In a sense, this is a very natural task for an octopus, which uses its arms to explore crevices etc. But normally, no eye-hand coordination is required, since crevices tend not to be transparent. The tentacle is therefore simultaneously sensor and actor and, presumably, relies mostly on its special tentacle-brain (the nervous system of the octopus is much more distributed than ours and each of the eight tentacles has its own dedicated neural apparatus). In contrast, in the study by Gutnick and colleagues, the octopus had to rely on its visual sense. The octopus was only given a single chance, so if it would simply probe around, like it would normally do, it would miss the food reward on two-thirds of the trials. The result of this study is beautiful in its simplicity: Yes, octopuses can learn this task. They see, reach and get it.
Let's finish with a nice clip from the documentary "Aliens from the Deep". (Here you can see footage of octopuses manipulating transparent objects as well, but the study by Gutnick and colleagues is new in the sense that the octopus wasn't allowed to "feel around" like the animals in the video.)
 There is some confusion about the correct plural form of “octopus”, but since the resemblance with the Latin “-us” suffix is superficial I would say that “octopi” is an ill informed attempt at sounding smart. You can also say “octopodes“, although I've never actually heard anyone use it. “Octopuses” it is, then.
 It must be noted that this finding is contested, although I fail to see any obvious methodological flaws myself. But you could argue that this finding is extraordinary, particularly given that octopuses are not social animals, and therefore requires stronger evidence than has been provided.
Fiorito, G., & Scotto, P. (1992). Observational learning in Octopus vulgaris. Science, 256(5056), 545-547.
Gutnick, T., Byrne, R. A., Hochner, B., & Kuba, M. (2011). Octopus vulgaris uses visual Information to determine the location of Its arm. Current biology, 21(6), 460-462.
Hochner, B. (2008). Octopuses. Current Biology, 18(19), R897-R898.