Zero-order reactions aren't common, but they're included with lots of chemistry courses because the math behind them is straightforward. Most examples of zero-order reactions are thermal decompositions - where a chemical just breaks apart because there's so much heat - because it isn't the reactant concentration controlling the reaction, it's the temperature!
The Rate Equation
Zero-order reactions mean the exponent on the reactant concentration is 0:
Notice how the whole rate expression simplifies to Rate = k. This type of reaction doesn't depend on reactant concentration!
How Concentration Changes with Time
Just like with first- and second-order reactions, you can integrate the rate equation to determine how the concentration of the reactant changes with time. If you know a little calculus (antiderivatives) you'll be able to follow the integration that I've put on the right. If you can't, just skip to the second-last line.
If the reaction is zero-order, a graph of [A] vs. t will give a line! Remember (from grade 9) how y=mx+b is the equation of a line? Well here, y is [A], the reactant concentration. x is t, the time elapsed. m (the slope) is -k and b (the y-intercept, where t=0) is [A]o, the initial reactant concentration.