Any period of ten years is a decade, including any arbitrary span of ten years; for example, the statement that "during his last decade, Mozart explored chromatic harmony to a degree rare at the time" merely refers to the last ten years of Mozart's life without regard to which calendar years are encompassed.
A popular and frequently referenced interval is based on the tens digit of a calendar year, as in using "the 1960s" to represent the decade from 1960 to 1969. Sometimes, only the tens part is mentioned (60s or sixties), although this may leave it uncertain which century is meant. These references are frequently used to encapsulate popular culture or other widespread phenomena that dominated such a decade.
However, the Gregorian calendar begins with the year 1. (There is no year "zero", and the year before AD 1 is 1 BC with nothing in between.) Therefore, its first decade is from AD 1 to AD 10, the second decade from AD 11 to AD 20, and so on. So, although "the 1960s" comprises the years 1960 to 1969, the years 1961-through-1970 comprise "the 197th decade" of the calendar, which can also be referenced as "the seventh decade of the 20th-century".
Particularly for the 20th century, a nominal decade is often used to refer not just to a set of ten years but rather to a period roughly approximating those ten years - for example, the phrase the sixties often refers to events that took place between c. 1964 and 1972, and to memories of the counterculture, flower power, protests of 1968 and other things happening at the time. Often, such a nominal decade will come to be known by a title, such as the "Swinging Sixties" (1960s), the "Warring Forties" (1940s) and the "Roaring Twenties" (1920s). This practice is occasionally also applied to decades of earlier centuries, for example, references to the 1890s as the "Gay Nineties" or "Naughty Nineties".
^Passim, i.a. Spencer, Donald D. 1989. Invitation to number theory with Pascal. Ormond Beach: Camelot. 46: "The first decade is from one to ten inclusive, the second decade from eleven to twenty inclusive, and so on."