When your teacher stays up until four in the morning assigning scores to student work, that's the grading of papers. Grading can also mean smoothing a surface, as in grading a lawn before planting seeds.
Grading comes from the verb grade, which basically means "group by category." When a teacher works on the grading of papers, she's dividing them into groups — the A's go together, the B's and so on. Fruit grading works in the same way, as does meat — you can have meat where the grading is "prime," or meat with a grading of "Grade A." Grading also has to do with the idea of gradual change — a steep slope could be said to have steep grading.