Every calendar in the world solves the same arithmetic problem, and there are only three solutions. A lunation is about 29.53 days and a tropical year about 365.2422 days, and the second is not a whole multiple of the first. A calendar can therefore follow the Moon and let its year drift against the sun, follow the sun and let its months lose contact with the Moon, or follow both and insert an extra month at intervals to keep them together.
The Gregorian calendar takes the second path and abandons the Moon entirely. Its months are conventional divisions of a solar year and have no relation to any lunation, which is why the word month and the word moon share a root that the calendar no longer honours.
It is a correction rather than a new system. The Julian calendar it replaced assumed a year of exactly 365.25 days, about eleven minutes too long, and the accumulated error had moved the equinox by ten days by the sixteenth century. The reform promulgated by Pope Gregory XIII in 1582 dropped those ten days and changed the leap rule so that century years are leap years only when divisible by 400, giving a mean year of 365.2425 days.
Adoption was slow and followed confessional lines, so European dates from the sixteenth to the twentieth century may be given in either style and are sometimes marked Old Style or New Style to say which. The calendar carries one lunar remnant: the date of Easter is still computed from a lunar cycle, and the computation of it was a principal motive for the reform.