First-order reactions are VERY common. The most common example is radioactive decay, where half of the sample disappears every 5 minutes (for example) goes by.
The Rate Equation
First-order reactions mean the exponent on the reactant concentration is 1:
How Concentration Changes with Time
Just like with zero- 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 fourth line.
If the reaction is first-order, a graph of ln [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 ln [A], the ln of the reactant concentration. x is t, the time elapsed. m (the slope) is -k and b (the y-intercept, where t=0) is ln [A]o, the ln of the initial reactant concentration.