Medical silver nitrate is typically purchased as matchstick applicators and is usually found in combination with potassium nitrate (75% silver nitrate and 25% potassium nitrate).
When this applicator is combined with water (moisture inside the nose or blood):
Nitric acid (HNO3) is produced along with an insoluble precipitate silver hydroxide (AgOH) which produces the characteristic brownish-white discoloration that can stain clothing as well as skin.
Nitric acid also known as aqua fortis and spirit of niter, is a highly corrosive strong mineral acid and is what initiates the biologic "cauterization" action. It is also what causes a patient to feel a burning sensation when applied.
Silver nitrate applicator sticks can occasionally be found on Amazon.com.