A natal chart is a piece of spherical geometry before it is anything else, and the mechanics are worth separating from the interpretation because they are not in dispute.
Three inputs are required: date, exact time, and place. From these the astrologer computes the positions of the Sun, Moon and planets along the ecliptic, the apparent path of the Sun through the year, expressed in degrees of the zodiac. Western practice uses the tropical zodiac, whose zero point is the position of the Sun at the March equinox, so the signs are twelve equal thirty-degree divisions of the seasonal year rather than the visible constellations of the same names.
The place and time then fix the houses, twelve divisions of the local sky measured from the ascendant, the degree rising on the eastern horizon at that moment. Because the ascendant moves roughly a degree every four minutes, an error of half an hour in a birth time changes the house structure materially, which is why astrologers ask for the time on the certificate and use rectification techniques when it is unavailable. There are several competing house systems, Placidus, Whole Sign, Koch, Equal and others, which divide the same sky differently and can put a planet in different houses on the same chart. Which to use is an unresolved argument within the practice rather than a settled matter.
Finally the aspects: the angular relationships between planets, measured in degrees along the ecliptic. Conjunction at zero, opposition at 180, square at 90, trine at 120, sextile at 60, each allowed an orb of a few degrees.
Astrologers read planet, sign, house and aspect together as a single statement. The chart itself is a computation; everything after it is interpretation.