A square number makes a square of dots, like 4 by 4. A cube number makes a cube of blocks, like 3 by 3 by 3.
32 means 3 × 3: a square with 3 on each side.
n2 (n squared) is n × n: a square of dots.
n3 (n cubed) is n × n × n: a cube of blocks.
Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
A number you get by multiplying a number by itself, like 4 × 4 = 16. That many dots make a perfect square. The first square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
It means 5 squared: 5 × 5, which is 25. It does not mean 5 × 2. In the same way, 5³ means 5 cubed: 5 × 5 × 5, which is 125.
A number you get by multiplying a number by itself three times, like 3 × 3 × 3 = 27. That many blocks make a cube: 3 layers of 9. The first cube numbers are 1, 8, 27, 64 and 125.
Children spot square numbers in the times tables at about 9 to 10, and learn the ² and ³ notation and cube numbers at about 10 to 11.