The rising sign (Aquarius or whatever) describes how a person appears to be, whereas the sun sign describes who they really are in a wider sense than just appearance. The charactersitics of the rising sign will somewhat modify and shape the characteristics of the sun sign.
To understand the rising sign, think of the appearance of that person in the world. At the moment of birth, what was on the horizon? The rising sign was appearing. Therefore the thinking is that the rising sign determines how the person appears to be, not exactly how they really are (i.e., sun sign).
Did you know that, given a person's natal chart, you can tell their time of birth just by looking for the position of the sun? If their birth chart shows the sun at 9 o'clock, then they were born at sunrise. If they happened to be born at the exact moment of sunrise, then their rising sign (aka their Ascendant) and their sun sign would be the same sign (such as Aries, Taurus etc.) as well as the same degree (5 degrees, 15 degrees or whatever). If their sun is located at 12 o'clock, then the person was born about noon. If it is at 3 o'clock, then they were born near sunset. If it is at 6 o'clock, they were born near midnight.
All the rising signs:
Today's Daily Horoscopes