Euler's Partition Theorem states that the number of ways to partition a positive integer into distinct parts is equal to the number of ways to partition it into odd parts. A partition of a number is a way of writing it as a sum of positive integers, where the order of addends does not matter. For example, the number 5 can be partitioned into distinct parts as 5, 4+1, and 3+2, totaling three distinct partitions. In contrast, it can also be partitioned into odd parts as 5, 3+1+1, and 1+1+1+1+1, which also totals three partitions, illustrating the theorem's equality.