A factor of a number divides it exactly; a multiple of a number is what you get by counting up in it. Every number has a finite list of factors and an endless list of multiples, so 'highest' goes with common factors and 'lowest' goes with common multiples — never the other way round. The highest common factor (HCF) is the biggest number that divides both, and it is what you want when something is being split into equal groups with nothing left over. The lowest common multiple (LCM) is the smallest number both divide into, and it is what you want when two repeating events have to line up again.
Worked example
Problem Find the highest common factor of 36 and 48.
List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
The factors in both lists are 1, 2, 3, 4, 6, 12, and the greatest of them is 12.
Tip Check it: 36 = 12 × 3 and 48 = 12 × 4. Because 3 and 4 share no common factor, 12 cannot be pushed any higher.