The thought is that all matter (whether solid, liquid, gas, simple, complex, etc) is comprised of atoms. Those atomic units bond together to form molecules, which in turn combine into more complex organisms to ultimately create all matter in our universe.
Similarly, interfaces are made up of smaller components. This means we can break entire interfaces down into fundamental building blocks and work up from there. That’s the basic gist of atomic design.
Interesting talk and article by the every entertaining Brad Frost. Apart from being fun to watch, Brad shows us an interesting way to approach design problems: divide them up into smaller pieces and do the same with those pieces, until you have every element of the site nailed down.
For small websites, building an entire library of “atoms” is probably not worth the time you’d have to put into it but for large and complex designs this would definitely be a time saver.