According to Utah Valley University, a positive Tollens' test means that the substance being tested contains aldehydes, as opposed to ketones. Aldehydes are easily oxidized, while ketones are not, which causes the differing reactions.
Wikipedia explains that the Tollens' test is also called the silver-mirror test in reference to what occurs when it is positive. Aldehydes cause the elemental silver to precipitate from the solution. Often, the silver covers the sides of the test tube, creating a highly reflective, mirror-like coating on the inside of the test tube. There is one exception to the Tollens' test. If the reagent is placed in a solution containing alpha-hydroxy ketones, mirror-like particles are formed.
Wikipedia explains that the reagent used in the Tollens' test is not produced commercially because it has a very short shelf life. The reagent is also used to place silver mirror finishes on mirrors and glassware. However, when it is used in such applications, it must be prepared in much larger quantities than are necessary for testing.
As a safety precaution, Wikipedia suggests that after Tollens' reagent is used, it should be mixed with dilute acid before disposal. This prevents the formation of silver nitride, which is dangerously explosive.
The Tollens' test and reagent are named after the German chemist who devised the procedure, Bernhard Tollens, according to Wikipedia.