The first citation of maunde to describe the Thursday before Easter in middle English comes in the mid-15C. It described not only The Last Supper in general but also the ceremony of the washing of the feet of the poor or downtrodden.
The immediate origin was Old French mandé. This in turn derived from the Latin mandatum or "commandment" (see mandate (n.)).
For Christians the crucial reference is to the opening words of the Latin church service for this day, Mandatum novum do vobis "A new commandment I give unto you" (John xiii:34). This new commandment is to love one another.
The supreme test of this commandment will, of course, be the events of the following day: Good Friday.