Counting all and counting on are names for strategies used by young children to perform basic arithmetic.
A young child who is asked to add 7 + 8 would first count the seven objects ("one-two-tree-four-five-six-seven"), then count the eight objects ("one-two-three-four-five-six-seven-eight"), and then count all of the objects all over again ("one-two-three-four-five-six-seven-eight-nine-ten-eleven-twelve-thirteen-fourteen-fifteen"). This strategy for adding numbers is called counting all.
After a long while of counting all, children progress to counting on. A child who uses this strategy to add 7 + 8 would first count the seven objects ("one-two-three-four-five-six-seven"), and then continue the count with the eight objects ("eight-nine-ten-eleven-twelve-thirteen-fourteen-fifteen"). This is somewhat more efficient than counting all.
After what can sometimes be a very long period of counting all and counting on, children internalize these sums and differences, become able to perform them without counting, and become able to apply these known facts to solving other problems in arithmetic.