Noncommutativity for transformations: when dressing up, you first put on socks, then shoes, when undressing, you have to take off socks shoes first!
Percentages and interest rates: it sort of matters whether to put 2,000 into your savings account this year and 1,000 the next year or 1,000 this year and 2,000 the next year (not obvious to many people).
Abstract maths can go far from "reality" into the "world of ideas": Banach-Tarski paradox.
And as to the questions like "why do you like maths so much?" I usually prefer to draw parallels between maths and arts/languages (not denying the applications either).