Afternoon is the period of the day after the noon and before the evening. The noon is a fairly easy thing to define, since it has an astronomical derivation, but when evening starts, and afternoon ends, is a more tricky matter.
There is not, as far as I know, a precise geophysical definition of afternoon, other than it includes the time between noon and sunset. If there were such a scientific definition, it would perhaps have little bearing on the cultural meaning of "afternoon". Part of this comes from the fact that while the earth has a deficient axis, the human cultures that live on it sometimes have schedules that pretend that it does not. For example, I live at the not-terribly far north latitude of 45 N, equal to Venice, Italy or Montreal, Quebec, and the sun may set after 9 PM in the summer and before 5 PM in the winter. In June, it may be 90 degrees Fahrenheit with the sun still shining at 8 PM, and that would be considered "evening", while in December, it might be 35 Fahrenheit, drizzly, and the sun may have set after a day when it never really rose, and that would be considered "afternoon".
In general, however, the "afternoon" derives its status from its physical properties. There is a limited amount of time in the day to work, and since work was traditionally based on physical tasks that required daylight, afternoon was traditionally the end of the work day. Even in an economy that has both fluorescent lighting and an increasing abstraction from actually producing anything, "afternoon" is the traditional end of the work day. In a social context, afternoon is often a time to do activities that require daylight, while still giving people enough time for a leisurely morning. Also, for college students, afternoon is a time to stumble out of bed and go back to playing video games.
I have heard rumors that in some lucky countries, afternoon is a time for sleeping. I can only imagine what life would be like if these rumors were true.