partial derivative is basically holding one variable constant and differentiating with respect to another variable.  for example: z = 3x + 4y

the partial of z with respect to x is 3 (hold y constant)
the partial of z with respect to y is 4 (hold x constant)

one use is in optimization problems.  if you want to find the minimum cost of constructing, oh i don't don't, say a can, you would probably use partial derivaties.

i'm sure someone in the field can give you a better use for it.