Best definition for distributive property of multiplication

Teach them FOIL (first outside inside last). $(a+b)(c+d)=\underbrace{ac}_{first}+\underbrace{ad}_{outer}+\underbrace{bc}_{inner}+\underbrace{bd}_{last}$. But then tell them that b is 0 here. So $a(c+d)=(a+0)(c+d)=ac+ad+0c+0d=ac+ad$. This is actually a high school Algebra II technique so it might be overkill.

Here is my definition.

$a\times (b+c)=a\times b+a\times c$

or from the right,

$(a+b)\times c=a\times c+b\times c$.

That's about it, you just need to remember to teach them both sides.

The distributive property of multiplication is that $a(x+y)=ax+ay$. If I were teaching children about this, then I would be reluctant to give this definition right off the bat. Instead, I would use examples. Why is it to so easy to calculate $2$ million plus $7$ million? Both of those numbers are very large, and yet we can compute the answer in a heartbeat. $2$ million plus $7$ million equals $9$ million for the same reason that $2$ apples plus $7$ apples equals $9$ apples. It's because we can add the numbers $7$ and $2$ together, and then give the units. In other words, $$ 7 \text{ million} + 2 \text{ million} = (7+2) \text{ million}= 9 \text{ million} \, . $$ The reason that we can do this is because of the distributive property, which says that $ax+ay=a(x+y)$. In the above example, $x=7$, $y=2$, and $a=\text{one million}$. There are other examples, too: since $7 \times 12 = 84$, and $9 \times 12 = 108$, $$ 16 \times 12 = (9+7) \times 12 = 12(9+7)=12(9)+12(7)=108+84=192 \, . $$ More complicated rules, such as the 'FOIL' method are a direct consequence of the distributive property: $$ (a+b)(x+y)=\underbrace{ax}_{\text{First}}+\underbrace{ay}_{\text{Outside}} + \underbrace{bx}_{\text{Inside}}+\underbrace{by}_{\text{Last}} $$ This method works because if we let $k=x+y$, then $$ (a+b)k=k(a+b)=ak+bk=a(x+y)+b(x+y)=ax+ay+bx+by \, . $$ Teaching students the FOIL method without even mentioning this reason is ill-advised in my opinion.