The commutative property could informally be called the "order doesn't matter" property. It's the property that tells us that adding two things to three things is the same as adding three things to two things (2 + 3 = 3 + 2) or that five groups of four is the same as four groups of five (5 x 4 = 4 x 5). Multiplication and addition both have this property, but subtraction and division do not. (4 - 3 ≠ 3 -4, 4 ÷ 2 ≠ 2 ÷ 4)
The associative property is the "grouping doesn't matter" property. It tells us that it doesn't matter if we do "one-and-two and then three" or "one and then two-and-three," we'll get the same result either way. It doesn't matter if we add 2 + 3 and then add 5, or if we add 3 + 5 and then add that to 2, we'll still end up with 10 in the end. Symbolically, we'd say that (2 + 3) + 5 = 2 + (3 + 5).
Finally, the identity property is the "mirror" property. Addition and multiplication both have a special number called an "identity element" - for addition it's 0, and for multiplication it's 1 - that acts like a kind of mirror for other numbers. If 15 asked itself "what number am I?" it could add 0 to itself to find out: 15 + 0 = 15 ("Oh, right, I'm 15!") It could also multiply itself by 1: 15 x 1 = 15 ("Yup, still 15!"). A little more formally, the identity property is the property that combining a number with the identity element under some operation like addition or multiplication leaves the number unchanged.
