In fact, the self does not exist as we experience it and relate to it. It too is impermanent, in a constant state of formation, and continually dependent on other factors.
What we experience as self is actually just input from our five senses (sight, hearing, touch, taste and smell), and the way the mind responds to this input with perception, feeling, and concept/thought. The central organizer of all of this activity is our Consciousness. In Buddhist terminology these are called the Five Aggregates.
These five factors (senses, perception, feeling, concept/thought, and consciousness) combine in a continuous mental ballet to create a sense of continuity - a story and our place in it - that feels like a solid thing in relation to the rest of the world.
Metaphors are often helpful in Buddhist explanations. The mental process that we think of as our reality is like a torch being twirled around so quickly that the light looks like a solid ring of fire. In fact, the circle of fire only has the appearance of solidity.
In the same way, our apparent reality (the interplay of the aggregates) is made of individual moments, but they move so fast that they seem to make a continuous, solid reality and self / ego. There is, however, never any actual continuity; no single entity that passes from moment to moment. This is Selflessness.