Regifting or regiving is the act of taking a gift that has been received and giving it to somebody else, sometimes in the guise of a new gift. One example of a formalization of this activity are the white elephant gift exchanges, in which items can be regifted from year to year.
In the USA, "National Regifting Day" is defined as December 18. In Canada, eBay marketed "National Re-gifting Week" as December 26-30 (December 26 being Boxing Day, a time traditionally associated with gift-giving).
Regiving differs from straightforward giving in that goods are not acquired specifically for donation. Typically, goods that have been received as a gift are offered to others, unbeknown to them that it was originally a gift to the person offering it. Often the motives are principally charitable but also includes giving items which are surplus to one's needs. However, re-gifting also refers the means of giving away unwanted gifts as a way of disposing them. Consider the "fruitcake gift" scenario. Someone receives a fruit cake but they don't like (or want) fruit cake but think someone else might. so give it away to them who also may not like fruit cake either. They re-gift the product, etc. and it ends in a potentially infinite loop of gift giving.
Regiving differs from recycling in that recycling is most often associated with breaking components down and rebuilding into new products.
The term was popularized by an episode of the NBC sitcom Seinfeld ("The Label Maker"), although the practice pre-dates the term substantially. In the episode, the character Elaine calls Dr. Tim Whatley a "regifter" after he gives Jerry Seinfeld a label-maker that was originally given to Whatley by Elaine.
Source: Wikipedia, "Regift", available under the CC-BY-SA License.