Kite's presentation is excellent, but I will add some to the picture.
Most astrologers when discussing this topic see two categories, Traditional and Modern. Traditional has subcategories, too. It is essentially a revival of some ancient techniques. Traditional may be Hellenistic or Medieval for instance, and there are other categories.
Modern astrology to most astrologers refers to the very popular form of astrology that became prevalent in the twentieth century. It is responsible for resurrecting astrology after it practically disappeared from view in the nineteenth century. Modern astrologers rejected some aspects of Traditional because they had little faith in some arcane practices. Most of the books available for learning astrology are of the Modern persuasion. Modern astrologers think that the fatalistic astrology that is the Traditional astrology tends to negate free will. Modern astrologers tend to think that Traditional astrology is too much like fortune-telling.
I have a different view. I see so-called Modern astrology as Contemporary astrology because it is not so very modern. So I tend to categorize with Traditional, Contemporary, and Modern. The Modern astrologers are not widely known to the general public. To check truly Modern astrology, look for Rheinhold Ebertin and Martin Seymour-Smith, for example.
True Modern astrology tends to discount houses and signs in favor of planetary contacts like aspects, configurations and planetary pictures which are based on midpoints between factors. Some Modern astrologers dismiss houses, and some do not.
I started with the Contemporary astrology which dominated the twentieth century, but I evolved to Modern astrology because it works far better for me in my practice.
Modern astrologers often see things that suggest fate, but they also promote free will. Modern astrologers, as opposed to Contemporary astrologers, often test their methods empirically to verify them. They are disinclined to doctrine or dogma.