Basically, they are numbers that are quick and easy to add, subtract, multiply, and divide without needing a calculator. For example 20 and 5 are compatible numbers because you can divide 20 by 5 in your head and come up with the number 4 - they don't require excessive thought or calculation.
Compatible numbers are numbers that look nice or friendly with each other when we do mental calculation to estimate a product, but especially a division
Examples: 400 and 10, 36 and 6, 2400 and 12, and 64 and 8 (all easy to do since they are multipliers of each other).
Hope that helps.