First, like celestial navigators, you have to imagine the earth sits still while everything revolves around it. It's much easier to understand this way. The zodiac is imagined as a circle surrounding the earth along the imagined path of the Sun moving around the earth. Astronomers call this the ecliptic. Every day the entire zodiac "makes" one full orbit around the earth. At the moment of birth, one of the 12 signs is on the eastern horizon of the place at birth That sign, and all the others move clockwise in relation to the earth. All of the 12 signs pass over the eastern horizon every day. So at the time of birth a sign is on the eastern horizon at the place of birth. That sign is your rising sign. You can be "an Aries" which means the Sun is in Aries when you were born, and you can have any of the 12 signs as a rising sign. It depends on where you were born and when you were born (time of day).
The same is true of the Moon, which is in one of the 12 signs at birth, and every other planet.
Rising simply means moving clockwise over the eastern horizon. It takes each sign roughly 2 hours to pass over the horizon. 12 signs x 2 hours = 24 hour day.
The astrological chart, designed by a "Mad Man from Dixie" according to Debbie Kempton Smith (see below) is the opposite of a compass. East is on your left, and South is at the top. So the signs move from the 9 o'clock position to the 12 o'clock position and so on. Therefore a sign on the Eastern Horizon "rises" to the noon position.
The rising sign is sometimes called the ascendant, abbreviated ASC. Each sign is made up of 30 degrees of arc. The degree of the sign on the eastern horizon is the ascendant or ascendant degree. The rising sign represents your physical body, and in traditional astrology it represents your moral character, intelligence (to a point) appearance, health, etc. It is YOU. Today because of Sun sign columns, dilettantes use the Sun for that. Sun sign columns are called "horoscopes," a bastardization of the term horoscope.
If you are interested in the topic, get a copy of Debbie Kempton Smith's book "Secrets from a Stargazer's Notebook." It's as good an introduction as any. There are precious few people on YA that know very much about the subject, and not one skeptic has any clue at all.