As you know, common Theists believe in a literal "God" who supposedly created and rules the universe; while common Atheists reject the idea of "God" altogether.
Generally,Spiritual Atheists are people who do not believe in a literal "God" (thus the term "Atheist"), but still consider themselves to be (often deeply) "Spiritual" people.
For Spiritual Atheists, being "Spiritual" generally means to have thoughts, words, and actions that are in harmony with the idea that we are all, in some way, connected - that the entire world can be thought of as one entity made up of smaller interacting entities.
Therefore, Spiritual Atheists generally feel that as they go about their lives striving to be personally healthy and happy, they should also be striving to help the world around them be healthy and happy. This empowering concept is referred to as "Wholistic Ethics".
Although Spiritual Atheists do not believe in a literal "God"; they generally recognize "God" as an important and powerful symbol of the inter-connectedness and one-ness of the Universe. They see that, individually, everything in the Universe is a part of "God"; and, collectively, everything in the Universe is "God". And that there is only ONE "God".
Even so, many Spiritual Atheists are extremely reluctant to make use of the word "God", due to the extreme desecration it has suffered by common Theists and Atheists alike.