Spirituality is the fundamental part of all of our religious and philosophical systems. It is the core of Christianity, Judaism, Islam, Hinduism, Buddhism and all other religions. It is also at the core of every thought system that we as humans have ever created: modern physics, communism, atheism, humanitarianism, and all the others too! This is because we, as humans, all have the same basic needs and wants. Like it or not, we all wear trousers or skirts, and eat a meal or two (if we're really lucky) each day. So, spirituality is concerned with our deepest and most important desires for happiness, peace, and a bit of fun.
The spiritual part of our nature is subtle and ordinarily quite difficult to fully experience. It is usually so deep down inside of the heart that, most of the time, one can barely hear it. The many experiences and sensations of mind, body, and the external world also tend to drown it out. Thus, spiritual practice is all about learning to get in touch with our most precious part, our spiritual nature.
Spirituality builds upon personal psychology and well-being. But it is not the same thing.