Although the Giant Squid (Architeuthis sp.) is already known to science, and thus is not technically a cryptid, there is still very much we do not know about the giant squid.
The history of the Giant Squid probably begins with the legends of the Kraken, which was discussed by Bishop Pontoppidan. The Kraken was a legendary monster that was believed to be as big as an island. When people landed on it (mistaking it for an island), the Kraken would sink, killing everyone on it. Descriptions of the Kraken, saying that it had many arms on it, have led some people to think that the Kraken may have been an exagerated account of the Giant Squid.
Interestingly, there are accounts of beached giant squids dating back to the 1600s. Because of the chemical makeup of the Giant Squid's skin, it floats after dying, giving it a chance of being washed onto shore. Sperm whales were also reported to have puked up large tentacles while the whale was being killed, sometime before the Giant Squid was accepted by science.
A Giant Squid was seen by the Alecton, a French warship, in 1861. They managed to get a rope around it, though the body slipped off, except for it's tail. A Giant Squid attacked a small boat in 1873. The crew of the boat managed to hack off some of it's tentacles and escape. One of the tentacles, which was 19 feet long, was obtained by Reverand Harvey, who also later obtained another carcass. The carcass was given to Professor Verill, who then classified it and settled the issue of the Giant Squid's existence.
Since then, many other Giant Squids have been washed ashore and caught in trawling nets. The Giant Squid is lives in the depths of the ocean, and if seen alive, is usually close to death.
There are a few accounts of Giant Squids attacking ships. In 1874, a Giant Squid pulled down a schooner, from which some of the crew managed to escape. During World War II (in 1941), after the Britannia had been sunk, 12 of the survivors were left on a small raft. At one point, one of the survivors was pulled down by a large squid, and (presumably) killed. Lieutenant R. E. G. Cox was grabbed by a tentacle, which let go and left a scar on him (the squid was probably about 23 feet long). In the early 1930s, the a Giant Squid attempted to attack a tanker, the Brunswick.
We have virtually no information about the Giant Squid's habitat (except that it lives in the ocean), it's prey (although Sperm Whales are known to eat Giant Squid), or how long it lives. The maximum size of the Giant Squid is also unknown. The largest authenticated Giant Squid was about 57 feet long, though there may be bigger squid. One reported sighting said that the squid was about 175 feet long. Some Sperm Whales was reported to have scars from Giant Squid that were about 10 cm in diameter. If these were the actual size of the Squid's tentacles, then the Squid would be about 80 feet long (it may also be possible that the scars grew larger as the whales aged).
Only a few people have actually seen living giant squid, and even then the Squid were most likely dying. There have been attempts to film Giant Squid in their own environments, by using submersibles. So far these attempts have been met with no success, and the Giant Squid still remains as one of the most elusive sea creatures accepted by science.