안녕하세요 켈다샘입니다. 

It's kind of hard to explain the exact meaning or perhaps to make students understand. 

But it is genuinely simple to answer. 

If you look at the rate law like Rate= k[NO]^2[O2], 

the reaction order with respect to NO is 2nd, and the reaction order with respect to O2 is 1st, right? 

The meaning of the reaction order is really simple, like

if you double the concentration of NO, the reaction will speed up by a factor of 4. 

If you double the concentration of O2, the reaction will speed up by a factor of 2. 

Yes, that's it. 

That's the only meaning and things you need to know. 

Reaction orders are not the SPEED by themselves. 

It doesn't mean NO is more quickly consumed. 

Rather, how the concentration of NO affects the reaction rate. 

We call it a reaction order. 

They are the coefficients of the reactants in each elementary steps, thus, they are deeply related to the reaction mechanism. 

Since mechanism is not proven, but only suggested, mechanism is researched to be consistent with those reaction order that comes from the experimental data. 

So why do we study kintetics?

That is to have a glimpse of idea to think of what's happening during the progress. (aka. mechanism) 

I hope it helps!