As with many of H.P. Lovecraft's creations, the Shoggoth is poorly defined. There is no accurate, complete description of the creature, nor any layout of the rules or behaviors it tends to follow or even a listing of its various abilities or vulnerabilities. It only appears in one book (At the Mountains of Madness, 1931) and was given a quick, fearful once-over by a pair of terrified characters running full-out to escape the thing.
Of course, this is Lovecraft's distinctive style, and he is both maligned and adored for it. While some complain that Lovecraft too often rips off the reader by describing things as "indescribable", others appreciate that this allows the reader's imagination to conjure something more horrible than any author could hope to put into text. A thing is all the more terrifying for what we cannot understand. For example, if one were to consider the case of the vampire, we see that the less that was understood about them, the more frightening they were. As they became more popular and appeared in more literature, given in-depth explanations as to their habits and abilities, and the nature of their vulnerabilities was explained too completely, they certainly lost a great deal of the edge they once held. In recent movies such as John Carpenter's Vampires and From Dusk Till Dawn, they seem downright silly.
Lovecraft's creations on the other hand are left vague, ambiguous, and without detail, and their ability to inspire horror has not been diminished in the decades since they were originally printed (save for a modern horror audience's overexposure to such things). The Shoggoth is certainly no exception to this rule. What the narrator knows about them appears to be pieced together from a combination of the Necronomicon and carvings found in the Old Ones' city. Its description is little more than this:
They were normally shapeless entities composed of a viscous jelly which looked like an agglutination of bubbles, and each averaged about fifteen feet in diameter when a sphere. They had, however, a constantly shifting shape and volume - throwing out temporary developments or forming apparent organs of sight, hearing, and speech in imitation of their masters, either spontaneously or according to suggestion.
These blob-like creatures were created and used as beasts of burden by the alien race of Old Ones (not to be confused with the Great Old Ones) who colonized Earth millions of years ago. They were able to shape their bodies into virtually any form, making them extremely versatile tools, and reproduced asexually by fission. Unfortunately, as they grew older they also seemed to grow more intelligent, possibly as a result of the hypnotic suggestions that the Old Ones used to control them. The Shoggoths at first became willful and difficult to control, and eventually (about one hundred and fifty million years ago) they rebelled outright, killing some of their former masters by tearing off their star-shaped heads. They were eventually resubjugated by the Old Ones, wielding strange molecular disruption weapons.
The mad author of the Necronomicon had nervously tried to swear that none had been bred on this planet, and that only drugged dreamers had even conceived them. Formless protoplasm able to mock and reflect all forms and organs and processes — viscous agglutinations of bubbling cells — rubbery fifteen-foot spheroids infinitely plastic and ductile — slaves of suggestion, builders of cities — more and more sullen, more and more intelligent, more and more amphibious, more and more imitative! Great God! What madness made even those blasphemous Old Ones willing to use and carve such things?
The Old Ones are now apparently extinct (at least on Earth, they may survive on other planets), and it would seem that the cause may have been a second Shoggoth rebellion. The narrator hints at this through sightings of headless Old One corpses littered around the fantastic Antarctic city they were exploring. Surely, if the Old Ones survived, they would have buried their dead, or performed their equivalent ceremony. Now the only inhabitants are the murderous Shoggoths, and how we would protect ourselves from an amorphous, cold-resistant blob able to assume any shape — should we ever encounter one — is left unexplored. The Shoggoths themselves appear to now survive in a grotesque, primitive parody of their former masters' society, carving rough approximations of the Old Ones' art and calling out "Tekeli-li! Tekeli-li!" in imitation of their voices.
Lovecraft's inventions have been somewhat expanded upon by various professional and amateur authors. According to these, Shoggoths tend to be summoned or created as multi-purpose servants by magic-wielding humans or other entities. They perform their tasks tirelessly and exactly to the whims of their controllers, and make powerful guardians because their amorphous shapes are very difficult to harm. Even having a pseudopod cut off is little more than an inconvenience, as it can be reabsorbed by the whole effortlessly. Any tool or limb required can be formed from the shapeless mass, including eyes, ears, teeth, tentacles, and weapons, solidifying as necessary to approximate steel or remaining nearly liquid. However, if owned for too long, the Shoggoth becomes increasingly willful and intelligent, and may become prone to lethal acts of rebellion.
Thanks to lusec for reminding me that Toy Vault (makers of the famous plush Cthulhu) also makes a plush Shoggoth! It's shaped like a starfish with multiple eyes and tentacles between the arms.