Democracy means the rule of the people (in Greek). That is where each individual person has a vote about what to do. Whatever the most people vote for wins. There is no king or tyrant, and anybody can propose a new law.
One problem that immediately comes up in a democracy is who is going to be able to vote. Should people vote who are just visiting from some other city-state? How about little kids, should they vote? Or should there be some limits?
The earliest democracy in the world began in Athens, in 510 BC. When democracy proved to be successful in Athens, many other city-states chose it for their government too. But most of them allowed even fewer people to vote than Athens did: most of the other city-states only allowed free adult male citizens to vote IF they owned land or owned their own houses (that is, the richer people).
Another problem for democracies was that it was very inconvenient for men to always be going to the meeting-place to vote. Most men had work to do, planting their grain, making shoes, fighting wars or whatever. They couldn't be always voting. So most democracies sooner or later ended up choosing a few men who would do most of the voting, and the rest only came when there was a really important vote. It was hard to decide how to choose these few men, and different cultures did it different ways. Athens did it by a lottery. If you got the winning ticket then you were on the Council of 500. Men served for a year.
Democracy spread around the Mediterranean, but it was pretty much wiped out by the Roman Empire about 100 BC. Still, places like Athens continued to use democratic methods to make their own decisions on local matters for a long time after
A thousand years later, in the Middle Ages, some cities in Italy - Siena, Florence, Genoa, Pisa, Venice - went back to having a democratic government. These were all organized in slightly different ways, but none of them allowed the poor or women to vote, and some had a lottery system like Athens.
(To read a description of the democracy of Florence written by a man who lived there at that time, click here).