The ascendant
The ascendant is the sign that was rising on the eastern horizon at the moment of birth. After the Sun it is the best known factor in a chart — and in interpretation it carries more weight.
How it is calculated
From the intersection of the ecliptic and the eastern horizon, for a particular place at a particular time. Three things are needed: date, exact time, place.
It moves about one degree every four minutes and runs through the whole zodiac in twenty-four hours. An hour's uncertainty in the birth time therefore shifts it by up to a whole sign.
That is why a serious reading asks for the time, and why newspaper astrology cannot use it.
What it means
The tradition reads it as the person: the body, the bearing, the manner of arriving in the world. The first house begins at it, and everything else depends on that.
Above all, the ruler of the ascendant — the planet governing the rising sign — becomes the most important planet in the chart. Its condition decides more than any single placement.
A Leo ascendant points to the Sun; dignified and angular is one thing, combust in the twelfth house is another. Both are "Leo rising", and they describe different people. "Angular" means on one of the four angles the ascendant forms with the descendant, the midheaven and the imum coeli; you can read what the other three mark in an overview of the angles and the nodal axis.
Why it says more than the sun sign
It distinguishes more finely. The Sun stays a month in a sign; everyone born that month shares it. The ascendant changes roughly every two hours — it separates people born on the same day.
It brings the houses. Without it there are no houses, and without houses one cannot say in which area of life a planet acts. That is half of the interpretation.
What it is not
Not a "mask", not "the image one projects". That reading is modern and narrows it.
In the tradition the ascendant is the person — body and life — not their front. The contrast the popular version builds, Sun for the inside and ascendant for the outside, has no counterpart in classical technique.
When the time is missing
Then the ascendant and houses are missing, and it is better to say so than to guess. The positions of the planets in the signs remain valid; everything that depends on houses does not.
There are methods for reconstructing a time — rectification, by the animodar for instance — but they are uncertain, and the temptation is to apply them until the result matches the expected reading. An honest uncertainty is worth more than an invented precision.
See also: the twelve houses, the planets.