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An attractive concrete description of this Boolean algebra is: the set of all periodic subsets of $\mathbb{Z}$, under the ordinary set-theoretic operations. – Nik Weaver Sep 19 '15 at 2:05

The 3-volume "Handbook of Boolean Algebras" abounds with examples of Boolean algebras, but of course only few of them are "nice".

  • You may have a look at Koppelberg's chapter 6 in volume 1, "Special classes of BAs", and also at several chapters in volume 3 describing some of these classes in more detail. This may challenge or sharpen your idea of "nice".
  • Generalizing your question one could also ask for nice constructions of Boolean algebras (products, free products, quotients, iterated quotients, interval algebras, completions, etc); some of them will yield nice algebras.
    For example, dividing any BA $B$ by the ideal $At(B)$ generated by the atoms will yield a new BA $B':=B/At(B)$; this new algebra $B'$ will often be atomless (e.g. if you start with a power set BA, as mentioned by Andreas Blass). If not, you can continue this process of dividing, even transfinitely often.
  • Monk's chapter 12 in volume 2 describes several methods for getting "many" BAs with particular properties. For example, starting with any linear order $L$, the interval algebra $I(L)$ is defined as the subalgebra of $P(L)$ (the power set algebra) generated by the half open intervals $[a,b)$ in $L$. Sufficiently different/nice linear orders will give nonisomorphic/nice BAs.
    For $L=\mathbb N$ you will get the (highly atomic) finite-cofinite subalgebra of $P(\mathbb N)$, and for $L=\mathbb Q$ you will get the countable atomless BA, mentioned by Joel Hamkins.
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An example of an atomic but not complete boolean algebra is the set of all finite unions of cartesian products of two sets (with the restriction that these sets are subsets of some fixed "universal" set, to eliminate set-theory paradoxes) with join and meet being set-theoretic union and intersection.

Atoms are cartesian products of one-element sets.

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