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The ascendant is the most personal point in a chart: the sign rising over the eastern horizon at your birth moment, in your birthplace. It changes signs roughly every two hours, which is why it needs your birth time — the Sun and Moon signs do not.
The ascendant is pure geometry. The Earth turns once a day, so the whole zodiac parades over the eastern horizon every 24 hours — about two hours per sign, faster or slower depending on the sign and your latitude (obliquely-rising signs take longer, which is why high-latitude charts skew). Given the instant and the place, the rising point is the intersection of the ecliptic with the horizon; this page computes it from the local sidereal time, and the formula behind it was checked against a brute-force scan of the horizon point by point.
Because the ascendant moves a degree about every four minutes, a birth time off by half an hour can move it seven degrees — usually within the same sign, but near a boundary it can flip the sign entirely. Birth-certificate times are rounded too; treat a result within ~4 degrees of a cusp as genuinely uncertain between two signs.
Yes, with the cusp warning in mind: if the result lands mid-sign, an error of an hour changes nothing; if it lands within a few degrees of a sign’s start or end, read both signs and see which description sounds like the person strangers meet.
In the tradition it is the chart’s anchor: it sets the houses, and it describes the native’s manner, appearance and approach — the interface between the person (Sun and Moon) and the world.
Identical — the Birth Chart calculator on this site uses the same formula with the same inputs, and shows the ascendant in place on the wheel.