The Death Spider is an enemy fought during the final boss battle in Carltron's Lair
Death Spiders spawn between attack waves, if the player kills the Mechadusters that come out to clean up enemy remains. The Spiders are very dangerous enemies with high stats, able to deal massive damage to the player with their attacks. The spawn from the same place as the Mechadusters, so the player should greet them with a charged attack to hopefully pierce their defenses.
A much simpler way to deal with them, though, is simply to leave the Mechadusters alone, as the Death Spiders will not spawn if the dusters are unharmed.
The player gets a good bit of experience from the spiders, so if they want to maximize levels before the rest of the final battle, killing them isn't bad idea.