(Chalky) You don't need to be under 30 to think of a spinster as unmarried and no chicken. The word was a bit old-fashioned even in my childhood (40's - 50's), and mildly derogatory. Its original meaning was "female spinner" (of yarn). This was presumably thought a proper way for unmarried ladies of a certain age to occupy their time, in the absence of vibrators, though who knows.
As Rosie says, a spinster is just an unmarried woman, but it has acquired unofficial overtones of left-on-the-shelfness. If you have banns read in church before your marriage, the woman is always described as "spinster of this (or another) parish".