Ernest Hemingway wrote his memoirs about his years in Paris in the 1920s, titling the book A Moveable Feast. The term far predates Hemingway’s use of it, though he no doubt encountered its actual meaning, especially in the strongly Catholic French nation. Hemingway uses the term to describe the constant movement and flair of Paris, instead of using it more accurately to refer to certain religious holidays.
A movable feast is more understood when you realize that feast day is often synonymous with holy day or holiday. In many religious, there are certain feast days that don’t fall on the same date yearly. In the Christian religion, for example, Easter is celebrated on different dates, and the formula for determining it is complex because it's not consistent. In Western Christianity, Easter tends to occur a week later than the Easter celebrated by Eastern Orthodox Christian churches. So not only is Easter a movable feast, occurring anywhere from early March to late April, there isn’t consensus on when it should be celebrated.
There are many Jewish holidays that are also movable feasts like Passover, which may sometimes coincide with the commemoration of the Christian Last Supper, which traditionally is described as taking place during a Passover feast. This varies too, and Jewish determination of any of its High Holy Days has nothing to do with Christian celebrations commemorating the life of Jesus. Chanukah is also a movable feast, with a specific number of days attributed to its celebration each year.
Contrasting to the movable feasts are those holidays that are fixed. Christmas, for example occurs on the same date year, 25 December. It's unclear why this is the established date for Christmas, since evidence suggests Jesus was born in the spring. Christianity may have chosen this specific date in order to replace previous holidays like Winter Solstice celebrations which occur at roughly the same time, thus making Christianity more palatable to the unconverted.
The definition of a movable feast also includes the statement that the feast must occur on the same day of the week, even if the date differs from year to year. This latter part is sometimes ignored when people define a flexible movable feast. One day that is not specifically a holy day, and is yet a movable feast is Thanksgiving. This is designated as the last Thursday of every November, but the actual date changes yearly. Further, Canada’s Thanksgiving differs in date from the United States.
In addition to movable feasts, there are many holidays that fall on the same day each year, but may not earn you time off of work when the holiday actually occurs. For example, Veterans Day is always the 11th of November, but for greater convenience, companies or schools may choose to shut down on the Monday before or the Friday after Veteran’s Day to give workers or students a three-day weekend.