Hume's impact on the theory of knowledge, ethics, and the philosophy of religion is enormous. Some of the issues he raised, for instance the problem of induction, remain as part of the philosophical problem-set. The power of his thought stems from a remarkably compact and incisive analysis of the human mind. Hume asks the reader to reflect on the contents of your own mind. You will find there many types of mental activity that you can group in various ways: perceptions, feelings, ideas, beliefs, emotions, expectations, doubts, and so on. Hume argued that all of these -- indeed everything that can be contained in the mind -- are reducible to two types of perceptions (any content of the mind of which we are conscious). These are impressions and ideas.
Perhaps you can perceive the taste of a plump juicy lemon right now. Imagine slicing the yellow skin, the juice spurts out, you put the wedge to your lips bite down, sucking out the juice. Does it work for you? Can you call up the lemon taste in your mind and almost feel the puckering of your lips in response to the sour juice? If so, you have focused on an idea, in Hume's sense of the word. That idea is in you because you have tasted actual lemons before. You have impressions of lemons in your mind and can make idea copies based on them; and (this is crucial) you can call an idea into consciousness without having the object (the lemon) of the original experience (tasting) present (on your tongue).
You can form an idea of something without being in its presence, but you cannot have an impression of something that is not present. Hume argues that ideas are actually copies of our impressions and that we can form complex ideas by combining simpler ideas. An argument against Hume's theory that all ideas (hence anything in the mind) come from impressions is to point out that humans are capable of forming ideas of things that they never have, perhaps never could, experience and have an impression of.
Hume counters this objection with a very important move. He points out how complex ideas may be formed by combinations of simpler ideas. For instance, you can right now form an idea of a unicorn - a horse with a horn. You probably have actually seen horses and seen horns in animals -- your ideas of horses and horns came from impressions (perceptions) of actual horses and actual horns. By putting the ideas together, you can create a complex idea of an animal that you have never perceived directly. Your mind has no impression of a unicorn, even though the idea can be constructed. More and more complex ideas are possible via this method of combination.
This move, is important because it introduces the main tool of Hume's thought - the ability to identify the relations between ideas.
If all we had to go on were impressions and ideas, we could not do much more than have perceptions and notice past experiences. Hume, however, develops a powerful account of the mind by identifying the ways in which ideas may be related to one another. We can mentally link ideas together in three ways:
Our mental processes are all made up of linking ideas together in these three ways. We have the ability to be aware of the ideas that we have and to be aware of the relations that hold them together. Hume argues that all human beliefs -- all of your beliefs -- result from applications of these simple associations. From our simple ideas and associations we build very complex system ns of thought and belief. yet, no matter how complex an idea or belief system, it is always possible in principle to analyze it into its simpler component parts -- ideas and relations between them.
This theory of mind and method of analysis provided the tools that Hume used to arrive at remarkable conclusions about knowledge, understanding, metaphysics, the self, morality, justice, religious belief, and a host of other key philosophical topics.