I am currently plotting density plots in R to view the distribution of my data.
An example density plot is here
I understand N is the number of values in the data set and bandwidth is the the smoothing used. But on the Y-axis what is the density? and how is it calculated?
The amount being plotted is an approximation to the probability density function of the population from which your data is drawn. If your data points are $(x_1, x_2, \ldots, x_n)$ then this is
$$ y = {1 \over n} \sum_{i=1}^n {1 \over \sigma} f\left({x-x_i \over \sigma}\right) $$
where $f$ is some nonnegative function with $\int_{-\infty}^\infty f(x) \: dx = 1$, called the kernel, and $\sigma$ is some constant related to what R calls the bandwidth. Essentially what this does is to put a peak of width approximately $\sigma$ at each data point and then average those.