There are several different notions of density in the integers. The most common is Natural Density. You can learn more at [wikipedia][1]. The idea is that you want to measure how large a subset of the integers is. One attempt is via [measure theory][2]. It turns out measure theory often works best in continuous settings, while the integers are discrete. For instance, you can't use the standard measure on $\mathbb{R}$ because the integers already have measure zero. There are plenty of other measures you can define on $\mathbb{Z}$ but it's not clear which is the best to capture number theoretic sizes like the "size" of the squares vs. all integers. Another attempt to measure size is via [Baire Category][3], and this works well in general topological spaces. Again, we're not picking up the number-theoretic information in the integers, and that's why natural density comes into play.


  [1]: http://en.wikipedia.org/wiki/Natural_density
  [2]: http://en.wikipedia.org/wiki/Measure_%2528mathematics%2529
  [3]: http://en.wikipedia.org/wiki/Baire_category_theorem