The sun represents your conscious mind. Your sun sign indicates your primary mode of consciousness, the basic way in which you experience waking life.
Your moon sign represents the unconscious. It indicates aspects of yourself that are fundamental, but that you may not be aware of. This is is also reflected in your dream life, and other altered states of consciousness, your presence on the astral plane, etc.
Your ascendent (rising) sign represents the manner in which your consciousness is expressed. It is your form. Outward personality, idiosyncratic mannerisms, etc are discernible through your rising sign. It is how others perceive you, to the extent that it can often indicate what you actually physically look like. The ascendent is often viewed as indicating your more "superficial" traits, although other schools of thought view it as being the most fundamental component of your chart, as it is the best indicator of your day-to-day personality. Another way I've heard it put is that your sun will show what it is you desire, whereas your ascendent shows how you go about getting what you desire.
Short of examining someone's full chart, I find that the sun, moon, and rising combined are the best way to get a sense of a person's personality. It's the bare minimum you need to make an assessment of someone astrologically.
Hope that was helpful. I'm by no means an expert on astrology, so to those more knowledgable than I, forgive me if I misspoke on any matter.
love is the whole and more than all