The six-month period during which the sun is said to be travelling southward in the solar year.
This is the time between the summer and winter solstices in the Gregorian calendar (approximately December 20 to June 20) and is based on the actual motion of the sun with regard to the earth.
The Indian calendar is based on the sun's movement around the zodiac, which is calculated differently from western astrology.
The Dakshinayana starts on Karka Sankranti (the day the sun is said to enter Cancer, which is generally July 14) and concludes the day before Makara Sankranti (the day the sun enters Capricorn, usually January 14).
Because the southern direction is connected with Yama, the deity of death, judgment, and retribution for evildoers, the Dakshinayana is considered less fortunate than the Uttarayana, the six months during which the sun moves north.
You may also want to read more about Hinduism here.
Be sure to check out my writings on religion here.