Jim sells his gold watch because he realizes that he cares more about making his wife, Della, happy than he does about keeping his most treasured possession. He knows that his love for her is more valuable than his watch, just as she realizes that her love for him is more important to her than her beautiful hair. This is why the narrator calls them the magi. The narrator says,

here I have lamely related to you the uneventful chronicle of two foolish children in a flat who most unwisely sacrificed for each other the greatest treasures of their house. But in a last word to the wise of these days let it be said that of all who give gifts these two were the wisest. Of all who give and receive gifts, such as they are wisest. Everywhere they are wisest. They are the magi.

The narrator contrasts Jim and Della with the original Magi, who brought "wise" gifts, but whose giving required no sacrifice on their parts. They were rich kings who could afford lavish gifts. Certainly, they gave gifts out of love, but Jim and Della's gift-giving required significant personal sacrifice on their parts; they had to realize that the happiness of their beloved was more important than their own, and it is this realization that makes them wiser than the three biblical wise men, who never had to make such a choice.

It is Christmas time, and Jim and Della are searching for Christmas presents for each other.  The problem is they do not have any excess money to buy presents.  They are on a tight budget. 

“Now, there were two possessions of the James Dillingham Youngs in which they both took a mighty pride.  One was Jim’s gold watch that had been his father’s and his grandfather’s.  The other was Della’s hair.” (pg 1)

Jim wants to buy Della some beautiful tortoise shell combs that she has admired in a store window on Broadway.

“….Beautiful combs, pure tortoise shell, with jeweled rims ---“(pg 2)

They were the perfect shade for her beautiful hair, and she really wanted them. Since he had no extra money, Jim sold the only thing he had of value, his watch.  The ironic thing is that Della sells her beautiful hair to buy Jim a chain for his watch, and Jim sells his watch to get the combs for her beautiful hair.  So neither of them ends up with a Christmas present.

In "The Gift of the Magi," when did Jim sell his watch?

The story does not specifically explain when Jim sells his watch.

However, Della goes shopping for Jim's present during the day of Christmas Eve. She finds and buys the watch chain, which indicates that Jim was still the owner of the watch when he left for his work that morning - Della would not have bought the gift if Jim no longer had the watch. Therefore, Jim must have sold the watch sometime during the day of Christmas Eve, giving him the funds he needed to purchase the hair combs as a present for Della.