Say that we're given a chemical reaction:
And the rate of the reaction is given by:
We say that the order of the reaction with respect to is and the order with respect to is . The total order of the reaction is given by .
One thing to note that and are determined experimentally. There's no way to predict the order without actually carrying out the experiment itself.
Now for specifics. A first-order reaction means that the rate of the reaction depends only on one of the reactants with order 1, i.e.
A common form is its integrated form: where is the initial concentration of our reactant.