Tonsils were an Andorian and Human organ located in the throat. Though part of the immune system, they often caused problems and were thus removed in a procedure known as a tonsillectomy. Tonsils could be examined with the aid of a tongue depressor.
Despite having access to the latest advances in medical technology, McCoy still believed there was a purpose in examining patients the old-fashioned way, and used these as an example. He was interrupted from examining Robert Crater's tonsils in 2266 by the apparent scream of his wife Nancy Crater. (TOS: "The Man Trap")
When James T. Kirk speculated that instrument malfunction was to blame for Doctor Leonard McCoy's abnormally healthy readings on the Omicron Ceti III colonists in 2267, McCoy stated the absence of his tonsils and his prior broken ribs being detected as proof of the medical tricorder's function. Later, he reported that, just as Kirk guessed, they had grown back, thanks to the effects of the Omicron spores. (TOS: "This Side of Paradise")