In American society, Christmas is a time when gifts are given to those a person loves. In the case of Jim and Della, the story doesn't explicitly tell us if this is their first Christmas as husband and wife but it implies that this is the case. Whatever the reason, Della has been thinking and dreaming of giving a fabulous, very special gift to reflect the extent and depth of her loving feelings for Jim. In the end, she achieves this dream by sacrificing the most valuable thing she owns - her hair - to buy a watch fob and chain worthy of Jim's pocket watch. The irony of the story is that Jim's present to Della, obtained by selling his heirloom pocket watch, is a set of hair combs to complement her long tresses.
The magi were the eastern kings who brought valuable but not immediately practical and useful gifts to the infant baby Jesus. In the same way, Jim and Della gave each other extravagant gifts paid for with significant personal sacrifice, only to discover that they were not going to be able to use the gifts any time soon.