Something that happens quarterly occurs four times per year, like your family's quarterly visit to your grandparents' house in Florida.
Magazines that are published every three months come out quarterly — in fact, they're commonly referred to as quarterlies. For businesses, many things happen on a quarterly schedule, from the quarterly taxes they pay to quarterly reports sent to shareholders. Think of a year as consisting of four quarters; anything occurring once per quarter happens quarterly. The Latin root, quartus, means "the fourth."