The reaction rate of any given reaction is the rate at which the components engage in the specific reaction, forming a new result (compound or precipitate, for example). The reaction order, on the other hand, is the coefficient applied to each component in the calculation of the reaction rate. The rate law is the mathematical expression of the rate of reaction, and this can take several forms: average rate over time, instantaneous rate at any specific point, and initial rate of reaction.

The reaction rate can remain steady or vary over time, and it can be affected by the concentrations of each component or by only one or two. Those concentrations can vary over time as the reaction continues so that the reaction rate is changing and the rate of change itself is changing. The reaction rate can also change based on other more obscure factors such as surface area available to the reagent, which can also change over time.

When the rate of reaction varies directly with the concentration of one component, it is said to be a first-order reaction. In lay terms, the size of the bonfire depends on how much wood you put on it. When the rate of reaction varies with the concentration of two components, it's a second-order reaction. Mathematically put, "the sum of the exponents in the rate law is equal to two."

When the rate of reaction does not vary depending on the concentration of any of the reagents at all, it's said to be a zero- or zeroth-order reaction. In that case, the rate of reaction for any specific reaction is simply equal to the rate constant, represented by k. A zero-order reaction is expressed in the form r = k, where r is the rate of reaction and k is the rate constant. When graphed against time, the line indicating the presence of the reagents goes down in a straight line, and the line indicating the presence of the product goes up in a straight line. The slope of the line varies with the specific reaction, but the rate of declension of A (where A is a component) is equal to the rate of increase of C (where C is the product).

Another more specific term is pseudo zero-order reaction because it's not a perfect model. When the concentration of one component becomes zero through the reaction itself, the reaction ceases. Just before that point, the rate behaves more like a typical first- or second-order reaction. It's an unusual but not uncommon case of kinetics, usually brought about through some artificial or otherwise atypical condition, such as an overwhelming preponderance of one component or, on the other side of the equation, an artificial scarcity of a different component. Think about a case in which a great deal of a certain component is present but not available for reaction because it presents a limited surface area for the reaction.

The rate law k has to be determined via experiment. Working out the rate of reaction is straightforward; it's real-world stuff, not algebra. If the concentration of the initial components decreases in a linear form with time or the concentration of the product increases linearly with time, then you have a zero-order reaction. If it does not, you have math to do.

Experimentally, you determine k using your initial concentrations or pressures of components, not the average, as the presence of the resulting product as time goes on can affect the rate of reaction. Then you rerun the experiment, changing the initial concentration of A or B, and observe the change, if any, in the resultant rate of production of C, the product. If there is no change, you have a zero-order reaction. If the rate varies directly with the concentration of A, you have a first-order reaction. If it varies with the square of A, you have a second-order reaction, and so on.

There's a good explainer video on YouTube.

With a little time in the lab, it will become obvious if you have a zeroth, first, second or more complicated rate law. Always use initial rates of components for your calculations, and within two or three variants (doubling and then tripling the pressure of a given component, for example), it will become clear what you're dealing with.