Hume argues that the self as traditionally understood doesn't exist. There isn't some stable, unchanging identity that remains in place throughout our various perceptions. In fact, what we call the self is actually just a collection, or bundle, of perceptions. Whenever we think of the self, we always think of it as doing something via the senses, whether it's seeing, touching, tasting, or hearing. At no point is it possible, except by abstraction, to get at a pure self separated from these actions. What we call the self is never isolated, it is always the self as this or that.
Hume's bundle theory stands in stark contrast to how Descartes construes the self. For Descartes argues that there is indeed a pure, disembodied self that lies behind all our perceptions. Essentially, Descartes turns Hume on his head by attempting to demonstrate that, whatever perceptions we have, there is always a self that accompanies them. In other words, for there to be any perceptions at all there needs to be a self doing the perceiving. Perceptions can't just float around freely; they need to be anchored in a perceiving self.
But Hume would challenge Descartes's notion of the self as an immutable substance underlying all our perceptions and judgments. To him, this savors of the kind of essentialism associated with medieval scholasticism.
Hume's critique of Descartes is radical for its day in that it dispenses entirely with all foundations in philosophy, whether it is God, or absolute morality, or the res cogitans, the mental substance which, in the thought of Descartes, underwrites all our perceptual experience of the world of time and space. By undermining the traditional understanding of the self, Hume hopes to dispense once and for all with what he regards as the unfortunate legacy of medieval philosophy and theology.