The invisibility device is a device which cloaks the wearer.
The cloaking device is a small device strapped around the arm. When the button on it is pressed, the wearer is shifted into an alternate dimension, which renders them invisible for people in the normal dimension. Only wearers of the same device or people who are affected by Arthur's Mantle can see the cloaked people.
The device gives off radiation which is harmful to humans over longterm exposure (Jaffa are protected from it through their symbiotes, and Ancients through their advanced biology), but it is necessary to prevent interdimensional parasites from bleeding through into the dimension humans occupy.
When the Sodan left the service of the Goa'uld, they settled on a world occupied earlier by the Ancients. There they found technologial instructions which allowed them to build the cloaking devices.
When the Sodan were wiped out by Volnek, who was turned into an undead monster by a Prior of the Ori, Teal'c used one of them to fight against Volnek and eventually defeat him. Afterward, the SGC collected all cloaking devices and brought them to Earth to study them. (SG1: "Arthur's Mantle")
Once the devices were analyzed, the radiation was detected and eventually disabled; however, when the SGC used them to spy on Ori-controlled worlds, they accidentally set free the parasites, which mutated animals into monsters. One cloaking device was stolen by a member of The Trust, who caused the parasites to be set free on Earth. All of the monsters were exterminated by members of the SGC. (SG1: "Uninvited")