A dasha is a planetary period. Vedic astrology divides life into chapters, each ruled by a planet, and the chapter you are in brings that planet's themes to the foreground. The most widely used system in India is Vimshottari, a cycle of 120 years.
The nine periods
The years are shared out like this: Ketu 7, Venus 20, Sun 6, Moon 10, Mars 7, Rahu 18, Jupiter 16, Saturn 19 and Mercury 17. They always run in that same order. Where in the cycle you start depends on the Moon's nakshatra at birth: each nakshatra has a ruling planet and that planet's period is the one running at your birth, partly used up.
Mahadasha and antardasha
The main period is the mahadasha. It is divided into nine sub-periods (antardashas or bhuktis) in the same order, beginning with the main planet, and each sub-period is as long as its share of 120 years scaled to the main period. A Venus mahadasha of 20 years, for example, contains a Venus-Venus sub-period, then Venus-Sun, Venus-Moon and so on.
How to read it
Start with the planet. What does it signify, and what does it rule in your chart: which houses it owns, and where it sits? A Jupiter period might bring teaching, children or guidance if Jupiter rules your 5th house, or travel and higher learning if it is tied to the 9th. Then read the sub-period lord the same way and look at how the two get along.
Dashas time events but do not create them. They are the weather, not the whole climate.
A caution
Avoid reading a dasha by the planet's reputation alone. A Saturn or Rahu period can be among the most productive of a person's life. Look at the whole chart.