Most often spiderwebs were placed inside bullets that were fired from projectile weapons. When the spiderweb entered an individual's body, it would begin to spread through the nervous system, inhibiting the transmission of nerve signals throughout the body. When the spiderweb reached the brain, it would halt all neurological activity in the brain, resulting in brain death. A spiderweb victim's eyes would cloud over and become silver in color due to the spiderweb spreading across their eyes. Spiderweb was always fatal, with no known treatment discovered by the mid 23rd century.
Fortunately spiderweb was rarely used as a weapon. The weapons were used in the mid 23rd century by a few terrorist organizations.
In an alternate timeline, Doctor Georges Mordreaux shot both Captain James T. Kirk and Lieutenant Commander Mandala Flynn with a spiderweb gun. Kirk died almost immediately. At first Flynn was able to move under her own power, and went to begin a search for Doctor Mordreaux before collapsing in a corridor and dying herself. Commander Spock was later able to prevent this timeline from occuring. (TOS novel: The Entropy Effect)