The only description given of it is that the hero should "shun the frumious Bandersnatch." Other advice given to the beamish hero proving quite accurate, one must dismiss the possibility that the speaker within the poem was uninformed about the Bandersnatch, which however does not actually make an appearance in "Jabberwocky". Thus it is clear that the Bandersnatch is not of a size or character to be embraced, perhaps putting it in a category not unlike rhinoceros. Elsewhere in Through the Looking-Glass, however, it is implied (but not stated) that a Bandersnatch is quick-moving, and that there may be more than one of it: "She runs so fearfully quick. You might as well try to catch a Bandersnatch!"
It is not stated whether there are not, necessarily, non-frumious Bandersnatches, but merely that the hero of the poem must shun a Bandersnatch that is frumious. At the same time, it may be that 'frumious' is not a merely descriptive adjective, but a definitive one, describing the essential quality of the Bandersnatch. It should be noted that a Google search of 'frumious' inevitably brings up 'Bandersnatch', and, conversely, no Bandersnatch can be googled without its frumiousness being mentioned.
In Larry Niven's fictional Known Space universe, the Bandersnatchi are colossal sluglike creatures. They were created by the Tnuctipun, used by the Thrintun as genetic engineers, as a food source by the Thrintun. Unknown to the Thrintun, the Bandersnatchi were created sentient, as spies, and immune to telepathy. They are also immune to radiation as their chromosomes are the thickness of a human thumb. Their sentience is largely related to the size of their massive brains, which were considered desirable by the Thrintun because of their delicious flavor.
The Bandersnatch also appears as an enemy in the video game Final Fantasy VII. It is a wolflike creature that can be encountered on the Northern continent.