I am trying to understand the concept of an atom in a Boolean algebra.
To fix the ideas, let $X=\{a,b,c\}$ be a set, and $\mathcal{A}=\{\emptyset,\{a\},\{b,c\},X\}$ be one of the five possible algebras of subsets of $X$. Of course, $\mathcal{A}$ being an algebra of set, it is also a Boolean algebra.
Now, an element $x\in\mathcal{A}$ in is a atom of $\mathcal{A}$ if, for every $y\in\mathcal{A}$, either $x\wedge y=x$ or $x\wedge y=0$. Am I correct to assert that $\{a\}$ and $\{b,c\}$ are the atoms of $\mathcal{A}$, and that, consequently, atoms in a Boolean algebra are not necessarily singleton sets (though every singleton set is an atom)?
In any partially ordered set with a minimum element, an atom is defined to be an element that covers the minimum element. The elements you refer to are atoms in this sense. This definition coincides with the one you give when the set is a finite lattice, as is a finite Boolean algebra.