Doctor Mortimer's walking stick is the first object that appears in The Hound of the Baskervilles. Holmes uses it to test Watson's skills at deduction, in which you make conclusions about something based on the general evidence you know to be true. Watson mostly fails at this game—though he's right that Doctor Mortimer practices medicine in the country, since his stick shows signs of lots of walking. However, nobody beats Holmes at this kind of stuff.
Holmes sees the inscription of "C.C.H." on the stick and figures out that Doctor Mortimer was a resident/student at the Charing Cross Hospital in London. From this fact, he reasons that Doctor Mortimer must be a youngish guy, maybe under thirty, who isn't too ambitious in his profession, since he's left London to practice medicine in the countryside.
Holmes observes the slight tooth marks on the stick and decides that Doctor Mortimer must have a medium-sized dog. All of Holmes' conclusions turn out to be right.
The walking stick episode in The Hound of the Baskervilles shows us right away—for anyone who isn't familiar with the Holmes stories—that these two are great friends. But still, Holmes is definitely the master and Watson is the student. Holmes is the great detective, and Watson is… not.
What's more, the walking stick gives readers the chance to see Holmes demonstrate his rational superpowers by showing off the kinds of conclusions he can draw from ordinary objects. To someone unfamiliar with Holmes, these conclusions could seem like a lot of fairy-tale-spinning. But Holmes insists that he has "methods" (1.25) that he applies scientifically to the stuff he observes, giving him greater insight than the unscientific observer. The stick represents everything you need to know about Holmes' powers of detection.