The Sun represents identity: what makes you say "I," your will, the way you express yourself and shine. The sign it occupies in your birth chart (your "sun sign," the one used in mainstream horoscopes) describes a style: how you assert who you are.
It is not the whole personality — the Moon (emotions) and the Ascendant (appearance, first impression) matter just as much, see Big Three. The Sun is the center around which the rest of the chart is organized, not the sole key to reading it.
Domicile: Leo. Exaltation: Aries. Detriment: Aquarius. Fall: Libra. The Sun is a diurnal planet — its sect strength (see the Aspects entry for the concept of sect) is at its peak in a chart born by day.
A planet very close to the Sun (less than ~8-17° depending on the planet) is called combust: its strength is judged weakened, "eclipsed" by solar proximity — except in a very tight conjunction (less than 0°17′), called cazimi, considered on the contrary a position of strength.
The Sun finds its joy in the 9th house (meaning, beliefs, long journeys).