Hamlet has it almost right but not quite.
New Years gifts were popular once, but these became obsolete by the early 19th century whereas the gifts given around Christmas survived. In the England of 1621, writers tell of dropping money at Christmas into an earthenware box kept an apprentice, which he'd break when it was full and buy himself a treat with the contents. By 1640 this custom had extended to servants in general. By the 1660s the practice had widened to making cash gifts from customers to tradespeople whose services they had enjoyed through the year. These cash gifts were now called "boxes" after the original practice. By the 1700s the upper-middle classes were starting to find the tradition oppressive as tradespeople were now demanding their boxes! The recipients didn't let it lapse without a struggle and in fact the custom flourished through the 1800s and 1900s. As recently as the mid 1960s, people actually feared the knock at the door of the refuse collectors expecting their Christmas box in case their rubbish was strewn around their garden if they refused! And when delivering newspapers, I was actually berated by customers who had wanted to give me a Christmas box because I hadn't knocked to collect it. In both these cases the box was a cash gift.
Official use of the term Boxing Day is less than 130 years old. When the British government passed an Act of Parliament in 1871 making the 26th of December a public (or "bank") holiday, it was still referred to by its old name St. Stephen's Day. Official secularization into Boxing Day is a late 19th century phenomenon.
If you want to read more about this an other customs, I fully recommend The Stations of the Sun by Ronald Hutton, professor of history as Bristol University.