Indian astrology
Indian astrology — Jyotish, “science of light” — is a tradition in its own right, practised without a break. It shares its Hellenistic roots with Western astrology, and the two have developed apart for almost two thousand years.
The main difference: the zodiac
Western astrology uses the tropical zodiac: 0° Aries is the spring equinox, the point where the celestial equator crosses the ecliptic. It is tied to the seasons.
Indian astrology uses the sidereal zodiac, which is set by the fixed stars.
Precession — the slow drift of the Earth's axis — pulls the two apart by about one degree every 72 years. Today the difference comes to some 24 degrees. In practice this means: anyone with the Sun in Aries by Western reckoning usually has it in Pisces by Indian reckoning.
Which zodiac is “right” cannot be settled — the two measure different things. The tropical one measures the relationship to the Sun, the sidereal one the relationship to the stars.
The lunar mansions
Besides the twelve signs, Jyotish knows 27 nakshatras — lunar mansions of 13°20' each. They correspond roughly to the daily travel of the Moon, and for the interpretation they matter at least as much as the signs.
The Western tradition knew lunar mansions too, but there they have fallen almost entirely out of use.
The timing techniques
Jyotish works with dashas — planetary periods that divide a life into stretches. The best known system, Vimshottari, rests on the position of the Moon in its nakshatra at birth and shares out 120 years among nine planets, with sub-periods and sub-sub-periods.
The Western tradition has a close relative in the firdaria, a technique of the same Persian origin.
And karma
The term belongs to the worldview in which Jyotish stands, not to its arithmetic. An Indian chart is worked out no differently just because it is interpreted within a framework that presupposes rebirth.
This is the point at which Western accounts regularly become inaccurate: they borrow the vocabulary and leave out the technique. Anyone with an interest in Jyotish should start with how it is calculated — the sidereal zodiac, the nakshatras, the dashas. The rest follows from there.
See also: the Western zodiac signs, Chinese astrology — a third system, different again.