For hundreds of years, mathematicians even as great as Leonhard Euler(1)
have tried to make sense of partition numbers, the basis for adding and counting. Progress has been made but there has never been a full theory to explain partitions. Answers have always led to more questions.
Basically, partitions are not considered by some to be part of number theory at all but to keep it simple for now, in number theory, a partition of a positive integer (n) is a way of writing n as a sum of positive integers.
Here is an example from the Classic Encyclopedia