After looking into this for a bit, I’ve discovered I’m woefully unqualified to answer this question :X
It appears to be a combination of things:
- Planets falling in their ruling signs
- Planets occupying one of the angular houses
- Planets forming numerous aspects with other important spots on your chart
How you process all that into a hierarchical list of dominates, however, I really just don’t know :\ Maybe someone else will be able to provide a more clear answer?
In a natal chart, the most critical elements are usually, in order of importance from highest to lowest (as I understand them):
- the rising sign of the chart (what house of the zodiac is on the Ascendant);
- the location of the ruling planet of that sign, if there no planet in that sign;
- the planet in the rising sign, if there is one;
- the position of the Sun;
- the position of the Moon;
- the position of Mercury;
- the position of Venus;
- the position of Mars;
- the position of Jupiter;
- the position of Saturn;
Every planet’s level of influence is judged by four factors called the Essential Dignities (in order of importance from most positive to most malefic) [and bearing in mind that this is ignoring the Mansions of the Moon, and the Decanates or Faces of the Zodiac]:
- Sign that it rules (domicile); (++)
- Exaltation (+)
- Triplicity (+-)
- Detriment (-)
- Fall (—)
There’s also a question of accidental dignities or detriments, namely what aspects each given planet has with the other planets in the sky. There’s very few right answers.
This is an ENORMOUS question because it touches on modern vs. “traditional” (i.e., 1500-1750 AD) astrology (vs. “medieval/Arabic” astrology, vs. Classical astrology, both of which are something else again).
In general, you should first track the Rising Sign followed by the location of the Ruling Planet of that sign; that will give you a sense of the person’s overall character; the Sun’s position will give a sense of what they’re famous for; and the Moon’s position will show you what their overall face is to the world. At that point, you can start drilling down into the positions of the specific planets.
I don’t bother with the three “new planets” of Neptune, Uranus and Pluto — astrologically, they haven’t been around long enough to know what they really do; while the innermost seven have been plotted long enough for centuries to get a sense of their function and purpose.