Handsel Monday (occasionally mis-spelled as "Hansel Monday") is an adopted tradition in Ireland - it mainly used to be celebrated in Scotland and northern England, making it across the Irish sea with immigrants and settlers. A minority tradition connected to a specific ethnicity, and slowly dying out. And it can be one of the Twelve Days of Christmas (or not as it is a kind of a movable feast).
When Is Handsel Monday?
Handsel Monday falls on the first Monday of the year, so the actual date in any given year can be anywhere between January 1st and January 7th. In 2014 January 6th, also known as "Little Christmas" or "Women's Christmas", was Handsel Monday - in 2016 it was January 4th (smack in the middle of the possible dates), in 2017 it will be January 2nd, with a Handsel Monday on January 1st or New Year's Day coming up in 2018.
Strangely enough there are (at least in some areas of Scotland, where the tradition is stronger) also celebrations of "Auld Hansel Monday", the first Monday after January 12, thereby adopting feast days according to the old Julian Calendar to dates in the new Gregorian Calendar - similar to the celebration of the Battle of the Boyne or the October Revolution in Russia.
What Is Handsel?
"Handsel" is believed to have its roots in the language of the Saxons (placing at least the name firmly in an English context), with a meaning of "delivering something into the hand". A due to be given. Not a massive tax - the meaning is nearer to a small gift, either money or goods, more as a token to ensure good luck from the beginning. Thus a handsel would be, generally speaking, a gift to mark (and bless) a beginning. By handing over this gift, the giver also avoids not only a social faux pas, but also a spell of bad luck.
In the 19th century Handsel Monday was the day it was already customary (reflecting an older folk tradition) to give both children and servants (both "dependants" of the paterfamilias, so to say) a present. Which should not be a sharp object - not for modern (and frequently simply maddening) Health and Safety reasons, but because it may "cut the ties". Similar gift-exclusions govern marriage gifts, to name but one other event. And one other important caveat was attached to gifts on Handsel Monday ... if you gave a purse, it should never be totally empty. Though in traditionally parsimonious (to be kind) County Cavan you'd be hard pressed to find more than a ha'penny or (today) a cent in the new purse, I guess.
Later, tips (money, more than anything else, was becoming important over time) were not only expected on the day by the servants of the house, but also by all those calling to the house on a regular basis - newspaper boys, the dustmen, the postman, delivery boys from the local butchers, bakers and possibly candlestick makers as well. Similar traditions are found in many places around the world - though like in Germany, for instance, much of this gift-giving would be either immediately before Christmas or around New Year's Day.
In the 21st century, Handsel Monday is all but forgotten in Ireland ... though in some families children are still handed small gifts on this day. Many modern Irish folks will be hard-pressed to even have heard about Handsel Monday at all and may roundly refute it, saying it's "not an Irish tradition". And they'd be correct, as mentioned above.
An interesting description of Handsel Monday proceedings in Limerick can be found in "The Park Danes" by Patsy Harrold:
"Hansel Monday" was also celebrated on the first Monday of the New Year. On that morning a young boy in each house would be wished a happy New Year and given a half crown hansel by his mother. The woman would then usher her son out through the back door of the house. After closing this door, the mother would open the front one and welcome the boy back into the kitchen. The son's wealth was short-lived, however, as the woman would quickly retrieve the hansel. Half-crowns were never too plentiful in Park.