The first record of Yom Kippur Kattan is in the Pri Chadash (Rabbi Chizkiya De-Saluha) to Orach Chaim 417. He attributes it as a custom from the Kabbalist known as the Ramak (Rabbi Moshe Cordovero - the generation before the Arizal). The original custom is primarily about fasting, with some additions of Slichos, etc. which developed.
Today, due to the general reduction in fasting, some may observe it without, but the primary idea is to fast. Unless someone started taking this upon themselves, they certainly don't have to fast.
Before Rosh Chodesh Cheshvan, Teves, Iyar and Tishrei (חטא"ת) there is no fasting. Cheshvan because it was just Yom Kippur, Teves because it is Chanuka, Iyar because there is no fasting in Nissan, and Tishrei because it is Erev Rosh Hashana.
The idea of it being "Yom Kippur" is that it is a day of forgiveness for the sins of the previous month (emphasized by the sin offering brought on Rosh Chodesh).