Astrology is the study of the stars and how they reflect our human condition. This is a broad statement, and there are many different variations of using the stars to determine what they tell us about people and places. What we commonly use in the United States is called Tropical astrology (often referred to as Western astrology). The beginning of Aries is determined by the moment each year of the Vernal Equinox (when the Sun crosses the equator going northward). Tropical astrology divides the sky up into 12 equal parts of the sky, each given a name (Aries, Taurus, etc.). The planets pass through these signs as they orbit the sun, and an astrologer analyzes the meaning of the location of these planets -- where these planets were found on any given day reflects a certain meaning.
Astrology is commonly used to interpret a natal chart (also known as a birth chart), which will reflect the basic nature, personality, and lessons of a person. But astrology can answer many other questions besides Who Am I? Here are some other uses for astrology:
Electional astrology is called for when someone wants to determine what the best day would be for a given event to take place. The most common use of Electional astrology is to determine when to have a wedding ceremony. Depending on your question, an astrologer will look at the pertinent planets that have to do with the nature of your event and looks at when their condition is most harmonious with the dates you have in mind. So, for example, if the event was a marriage, an astrologer would most likely take into account the condition of Venus, for one thing, which is most readily associated with weddings and relationships.
Mundane astrology is the astrology of places. Your town, your country, etc. all have a 'birth' chart, signified by the moment in time that the place is considered to be 'born'. Analyzing this chart can give insight into the vibe that a place carries and how it, as an entity, behaves. Determining the time of birth of a place can be rather tricky, since there are usually many significant moments in a place's history that lead up to its birth/creation. Does that mean it's the moment documents were signed? Wars were won? Not everyone will agree.
Astrocartography is also an astrology of places, after a fashion, but this has more to do with your birth chart and how it interacts with places, rather than just the birth chart of any given place itself. Astrocartography is a technique that maps out your planetary lines all over the world (through a method too complex to discuss fully here). Each planet in your chart lines up with a certain area of the world, and the idea is that if you were to visit these places, you would have experiences that are of the nature of that planet. If your Jupiter line runs close to Paris, for example, you may have a lot of Jupiterian experiences were you to go to that city, such as more opportunities seeming to open up for you, and experience a more light-hearted and optimistic attitude while you were there.
Relocational astrology takes your birth chart and relocates it from your birth place to any city in question that you want to compare. By then reanalyzing your natal chart according to the new place, you can determine how that place may influence your personality and what sorts of lessons and opportunities may arise in your life if you were to relocate to that city.
Horary astrology is a chart cast in answer to a question. The chart is cast for the moment the question is asked, and by reading it, an astrologer can address the question and come to an understanding of the outcome or answer. Much is based on the skill of the astrologer and the consciousness of the querant at the time the chart is cast, it is by no means a simple task.
Medical astrology is based on the idea that a chart can tell you not just about your figurative insides, but can be read entirely with your literal insides (and outside) in mind as well. As a very broad example, Leo is associated with the heart and the upper back. Knowing a person's chart can aid in understanding what ails them, how they are physically vulnerable to illness and of what sort.
There are also forms of astrology that are not as much used for different purposes but actually based on entirely different systems than the one commonly known in the U.S., which is Tropical/Western astrology. A few examples:
Chinese astrology is based on the Chinese zodiac, which is made up of different animals than the Tropical zodiac. While the Tropical system uses 4 elements, the Chinese uses 5. It is also more complex than it first appears, just as Western (Tropical) Astrology is. It utilizes the principle of the four pillars, which hold different insights for the year, month, day, and hour of your birth.
Vedic astrology (Jyotish) uses the sidereal system to calculate the planetary positions, but that is not the only difference from Tropical astrology. The interpretations of the planets, signs, and houses also have many differences, and even the charts are drawn differently -- while Western astrology typically uses a wheel to represent a chart, the Vedic system uses a square.
Sidereal astrology is used in the Vedic system of astrology. It is based not on the signs (the section of sky known as Aries or Taurus, etc.) but on the actual constellations (the stars that make up the constellations of Aries or Taurus, etc.). The signs and constellations are not the same thing. While Tropical astrology begins measuring the signs according to the moment of the Vernal Equinox, Sidereal astrology begins measuring the signs according to the location of the constellations. The constellations themselves are of varying sizes and don't take up equal sections of the sky like the signs do. Due to irregularities in the Earth's orbit, the constellations don't necessarily line up with the signs of the same name as they once did. Therefore, one could argue that Aries starts where the constellation does (Sidereal) or when the Vernal Equinox takes place (Tropical).