Second-order reactions are VERY common. There are two types of second-order reactions ... the simplest (the kind in most chemistry courses) is when two molecules of A collide together to form a new product. Another kind (not covered in first-year chemistry classes but in upper-year ones) is when two different molecules (A and B) collide to form new product(s).
The Rate Equation
Second-order reactions mean the exponent on the reactant concentration is 2:
How Concentration Changes with Time
Just like with zero- and first-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 last line.
If the reaction is first-order, a graph of 1/[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 1/[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 1/[A]o, the ln of the initial reactant concentration.