Astrology and astronomy both involve the study of the stars and planets, and, increasingly, the search for elements of the meaning of human life in space. Historically, they were even considered essentially the same process. However, today astronomy and astrology tend to refer to quite different activities: astronomy is the scientific study of space, and astrology is a set of spiritual beliefs about the metaphysical relationship between bodies in space and human life on Earth.
Astronomy is derived from the Greek for "star laws." It involves the study of planets, stars (including our own Sun), asteroids and comets, galaxies, nebulae, and so on. The field is actually extremely broad, since it can technically refer to the study of anything beyond the atmosphere of the Earth and therefore touches on elements of physics, chemistry, meteorology, and (where the search for extraterrestrial life is concerned) biology and even social science. Historically the field was largely limited to stargazing in the visual spectrum, either with the naked eye or (after the European Renaissance) a telescope. Today, though, astronomy has a much wider skill set, and regularly scans the skies in terms of radio waves, ultraviolet light, X-rays, microwaves, and gamma rays. The purpose of astronomy is to interpret and understand what it sees in natural terms: that is, to understand the nature and composition of what we see in the night sky, to understand how it came to be as it is now, and also to speculate about the natural evolution of these objects in the future.
Astrology, in contrast, is derived from the Greek for "star study." There are a number of extremely comprehensive astrological traditions still in use today, including those drawn from Hindu Indian, ancient and medieval Europe, and China. In general, astrologers attempt to connect an individual's life to the movements of the celestial bodies (i.e. planets), and from that information to determine information about personality and destiny. Although the practice has no specifically scientific merit, it is still very popular, including in America, where about one-third of people say they believe in at least some elements of astrology.
Until the European Enlightenment of the 17th-18th centuries, and even during much of that time, there was minimal distinction drawn between what we now refer to as astrology and astronomer. This was because cultures understood what they saw in the night sky in terms of their cultural and religious beliefs. The distinction between understanding and predicting the nature of the stars, and interpreting the religious and social significance of those stars, was razor-thin. Many of those now recognized as early heliocentric astronomers, such as Tycho Brahe and even Galileo, would have considered themselves astrologers as much as astronomers. Even today, both still share the fundamental assumption that the Earth and the rest of the universe are fundamentally interconnected, even though they approach that assumption from different directions.