A mole is defined as 6.022 x 10^23 units of any given substance. This number is also called Avogadro's number. It is used extensively in chemistry to quantify numbers of atoms of elements or chemicals in easier to use numbers.
Mole day is a chemistry celebration that started some time in the 1980's. It is recognized as occurring at 6:02 either AM or PM on October 23. Since October is the tenth month of the year, that makes the date 6:02 10/23, which closely approximates Avogadro's number.
Different groups do different things for Mole day. Some people write poetry about it. I have read about one chemistry couple that made it their wedding day. I don't want to copy and paste a poem that someone else wrote verbatim, so I will simply provide a link below to a site that lists 3 example Mole day poems.