Ignore the stuff about cannonballs. That is simply intended as motivation for introducing the equation y^2 = x(x+1)(2x+1)/6. If x and y represent numbers of cannonballs then of course they must be integers. But in the equation y^2 = x(x+1)(2x+1)/6, x and y are to be thought of as real numbers (maybe the book doesn't emphasise this transition from whole numbers to real numbers). So you should think of the equation as representing the graph of a variable ; in other words, a curve.
If you want to know why it's an elliptic curve, you'll find the definition here.