Converging sieves: Let $q_i$ be a sequence of integers with $\sum\frac{1}{q_i}$$\sum\frac{1}{q_i}<\infty$, and pick for each $i$ an integer $a_i$ within a finite set $\mathcal{A}$. Then the set of integers $n$ such that $n\not\equiv a_i\pmod{q_i}$ has positive density. Most examples involve coprime integers $q_i$, an example where the $q_i$ have large common divisors is the set of integers $n$ such that every group of order $n$ is solvable.
Sets of integers defined by values of arithmetic functions: For every $c\in(0,1)$ the set $\{n:\varphi(n)<cn\}$ has positive density, and for every $c>1$ the set $\{n:\sigma(n)<cn\}$ has positive density. Many other examples of this type arise from the Erdos-Kac-theorem.
Logical limit laws: If $\varphi$ is a statement in the unary second order language of groups, then the set of all $n$, such that all abelian groups of order $n$ satisfy $\varphi$ has a Dirichlet density, which is quite often strictly between 0 and 1. Other examples can be deduced from the examples in Burris' book "Number theoretic densities and logical limit laws".