The "rising sign" or "ascendant" is the zodiac sign sitting on the eastern horizon at the moment and place you were born. That is where the name comes from: the constellation is rising up over the eastern horizon.
The ascendant is an important idea in astrology. Some astrologers think of it as how you present yourself to the world; others as the lens through which you look at the world.
The rising sign changes every two hours
Every day the whole zodiac appears to make one lap of the sky. So every two hours a new one of the twelve signs rises from below the horizon. It is not really the signs that move but the Earth itself: it turns 360° every 24 hours.
- The Earth orbits the Sun (one lap = one year). Planet signs (e.g. ☿︎ Mercury in Aquarius) are based on this motion.
- The Earth spins on its own axis (one turn = one day). The rising sign is based on this motion.
Two people born at the same moment, on opposite sides of the Earth
Their charts are nearly identical, because seen from Earth the planets are in front of the same constellations. The only thing that differs is their horizon: what they see when they look east. So the only thing that differs is the rising sign.