In tropical astrology, the most commonly used astrology in the West, each sign has an equal 30° portion of the sky. The constellations are no longer located in their respected sky segment, because the Earth wobbles on its axis: When the Babylonians were establishing the principles of astrology thousands of years ago, the constellations and the signs of the zodiac were all in alignment. On the vernal equinox, at that time, the Sun was in the constellation Aries, which is why Aries is the first of the zodiac. Due to the shifting axis of the Earth, the constellations are in retrograde. -- That is, they appear to move backwards through the sky.
When the Sun is located in a constellation on the vernal equinox, it is the age of that zodiac sign. So, when astrology "began," it was the Age of Aries. Each age is said to last for approximately two-thousand-one-hundred-sixty years (it moves one degree about every seventy-two years: 72 years x 30° = 2160). We are said to be in the Age of Pisces, possibly entering the Age of Aquarius (no one really seems to be able to determine this). Some astrologers believe that Ages affect mankind, while others believe the Ages correlate with the rise and fall of powerful civilisations and cultural tendencies.
The star signs are the zodiac signs themselves. The luminaries are the heavenly bodies, such as the planets, the Sun, Moon, and various asteroids, &c. And the aspects are the relationships between the luminaries.
Each individual has multiple signs, one for each luminary. It works just the same as the Sun sign, except, instead of the Sun, the other luminary is in its place. For example, if the Moon is in front of the sky segment of Aries when you are born, then your Moon sign is Aries. Another way to say that is, "At the time of your birth, the Moon was in Aries; therefore, Aries is your Moon sign." Each of these signs represents a different aspect of your personality.
As a side note, I'd like to add that I recently read that some people believe astrologers are hiding that the signs are actually 30° segments of the sky, rather than actual constellations. This is not true. As astrology has always had to do with 30° segments of the sky, anyone that has taken the time to learn about astrology will have learned very quickly that it was never really about constellations, and that they were always only representations.
And please also note that the zodiac signs in newspapers and magazines are just for fun and should be taken lightly.