It’s hardly within my area of expertise to provide you with a concise and entirely accurate introduction to the history of Maundy Thursday, and the role that coins have played in it (in the United Kingdom), so be advised that the information that follows has been entirely collated from information that’s readily available online - click any of the links for more information about the subjects discussed.
Many sources describe Maundy Thursday as the Christian holy day just before Good Friday, and that it commemorates the Last Supper of Jesus Christ with the Apostles. The term “maundy” refers to the foot-washing ceremony that Jesus Christ originally performed on his disciples.
While a foot-washing ceremony will strike many non-Christians in the 21st century as being rather archaic, this passage of the Bible shows that Christ intended it to be a physical demonstration of the equality of all in the eyes of God - no man is so higher or lower than another that they cannot wash their brother’s feet, or have them washed:
"If I then, your Lord and Teacher, have washed your feet, you also ought to wash one another’s feet. For I have given you an example, that you should do as I have done to you. Most assuredly, I say to you, a servant is not greater than his master; nor is he who is sent greater than he who sent him. If you know these things, blessed are you if you do them."
There are many countries in the world where many churches of the Christian faith observe Maundy Thursday as part of Easter activities - Great Britain however is the only country in the world that has included coins as part of the commemorations.
According to the Royal Mint, the tradition of maundy “money” in Great Britain is thought to have started “as early as the thirteenth century”, when “…members of the royal family to take part in Maundy ceremonies, to distribute money and gifts, and to recall Christ's simple act of humility by washing the feet of the poor.”
King “…Henry IV began the practice of relating the number of recipients of gifts to the sovereign's age, and as it became the custom of the sovereign to perform the ceremony, the event became known as the Royal Maundy.”
“In the eighteenth century the act of washing the feet of the poor was discontinued, and in the nineteenth century money allowances were substituted for the various gifts of food and clothing.”
“Maundy money as such started in the reign of Charles II, with an undated issue of hammered coins in 1662. The coins were a four penny, three penny, two penny and one penny piece but it was not until 1670 that a dated set of all four coins appeared. Prior to this, ordinary coinage was used for Maundy gifts, silver pennies alone being used by the Tudors and Stuarts for the ceremony.”
“Maundy money has remained in much the same form since 1670, and the coins used for the Maundy ceremony have traditionally been struck in sterling silver…”
“Today's recipients of Royal Maundy, as many elderly men and women as there are years in the sovereign's age, are chosen because of the Christian service they have given to the Church and community. At the ceremony which takes place annually on Maundy Thursday, the sovereign hands to each recipient two small leather string purses. One, a red purse, contains – in ordinary coinage – money in lieu of food and clothing; the other, a white purse, contains silver Maundy coins consisting of the same number of pence as the years of the sovereign's age.”
In 2015, the Queen performed the pre-Easter tradition of handing out commemorative coins to the elderly in Sheffield. The Queen handed out two purses – one red and one white to each of the chosen recipients. The white purse contained Maundy coins to the value of 89 pence, while the red purse contained a £5 and a 50p coin.