The Treatise is Hume's attempt to put a complete science of human nature on a new footing, built on experience and observation in the way natural philosophy had been. He argues that all the sciences depend in some degree on the study of the mind, so understanding human nature is the foundation for everything else. The method is deliberately empirical: watch what the mind actually does rather than assume what it must do.
Book I, on the understanding, begins with the raw material of thought. Hume splits all perceptions into impressions, which strike with force, and ideas, which are their fainter copies. His governing principle is that every simple idea is derived from a simple impression that resembles it. From this he can challenge any concept by asking which impression it came from, and concepts that fail this test are treated as confused or empty.
The central case is causation. When we say one thing causes another, all we ever observe is contiguity, succession, and constant conjunction: the two have always appeared together. We never perceive a power or necessary connection in the objects themselves. What we call necessity is really a habit of the mind, formed by repetition, that carries us from the cause to expect its usual effect. Belief in matters of fact rests on custom rather than on demonstrative reasoning.
Turning to the self, Hume applies the same test and finds no impression of a constant, simple ego. Entering most intimately into himself he meets only particular perceptions of heat or cold, pleasure or pain, never a self apart from them. So the mind, for most of us, is a bundle of perceptions in continual flux, and the sense of a single enduring identity is a fiction the imagination builds from the resemblance and succession of those perceptions.
Books II and III carry the inquiry into the passions and morals. Hume holds that reason by itself never produces action; it only serves the passions by tracing the means to ends we already desire, which is why he calls reason the slave of the passions. Morality follows the same pattern. Examine a vicious act and you find no quality of vice in the facts themselves, only a sentiment of disapproval that arises in you. Virtue and vice are felt, not reasoned out, and Hume even notes that writers slide without warning from claims about what is to claims about what ought to be.