Theist/Atheist: Those that are convinced or unconvinced in the existence of a God(s)
Agnostic: Those who are uncertain to the existence/nonexistence of a God
(Non)Spiritual: Someone who has some sort of innately spiritual belief such as the existence of a Soul.

You can be one, or all of the above, but you can't be a Theist and an Atheist at the same time, naturally. You can be an Agnostic Theist, someone who is unsure as the the existence of a God but still believes there is despite this uncertainty. An Agnostic Atheist, someone who is unsure to the existence of a God and, usually due to that, does not believe there is one. A Spiritual Agnostic Atheist, someone who fulfills the above criteria but is also convinced to the existence of Angels and Demons, just as an example. This is not a comprehensive list of examples and more variables exist, such as what could be referred to as a Gnostic (not the Christian denomination) someone who is convinced of the existence or at least the possibility or probability of the existence of a God(s).

Atheism and agnosticism are pretty straight forward but spirituality is far more ambiguous. I believe I'm spiritual because I can be so emotionally moved by things (real things that is) For example, I can be brought to tears when I hear a choir sing Handel's messiah. I'm certainly not religious but it is an amazing piece of work. The exuberance of standing on a precipice, looking down on the vast valley bellow. The thrill of riding my motorbike far too fast, weaving in out of city traffic. THe awe of looking at the vastness of stars as I reflect on how insignificant I am in comparison to the universe. When I create something beautiful and original, it is like an extension of me. When I connect with someone who is a stranger and we realise we are so like-minded. I believe in the interconnectedness of humans and all living things and that no one should be alone because we need each other to thrive. I see the beauty and the pain and this to me is a spiritual experience.

I like Matt Dillhanty (spelling?) definition. Atheist is a lack of belief in a god as there as never been any proof of the existence of a god.
Agnostic is much the same. The word gnostic means knowledge and agnostic means the lack of knowledge of a god or gods.
Spiritual is harder to define as that is a personal experience. For me it is wonder, awe, joy I feel when I contemplate the Cosmos.

Atheist say there is no God (which is a belief because they they don't really know that). Agnostic says they just don't know (which is the correct choice). Spiritual means increasing one's feeling of self awareness while we are here on a search for meaning in our lives. Trying to be spiritual is a noble undertaking.
However we should be careful not to equate spiritual with "soul" because more than likely the is no such thing as a soul.