The sign climbing over the eastern horizon at the moment you were born.
Also called the Ascendant. Stand where you were born, at the time you were born, and face east: the rising sign is the constellation coming up over the horizon. It is called rising because it is, at that moment, ascending into the sky.
Because the Earth turns once a day, the whole zodiac files past the eastern horizon every twenty-four hours. A new sign begins rising roughly every two hours. This is the fastest-moving thing in your chart by a wide margin, and the reason an accurate birth time is worth chasing down.
Two rotations, two kinds of placement
Two motions are running at once. The Earth orbits the Sun, once a year: that gives you your planetary signs. The Earth also spins on its axis, once a day, and that gives you your rising sign and your houses.
Two babies born at the same instant on opposite sides of the world have essentially the same planets, because from anywhere on Earth the planets sit in front of the same distant constellations. What differs is the horizon: what is coming up in one place is going down in the other. Only the rising sign and the houses change, but they change completely.