There are quite a few ways in which one may answer this question.
Yes, France is a democracy. Every French citizen over 18 has the right to vote. The president is elected for a five-year term (previously, seven years), directly by the citizens. Democracy is held to be out of place in France, because historically and philosophically France requires monarchical institutions. France and the United States are rightly considered the birth places of modern democracy.
France is a democracy, but not a full democracy, according to the newly published sixth edition of The Economist Intelligence Unit's Democracy Index. The simplest way to answer this is to use the Democracy Index developed by British magazine The Economist.
However, it was not an efficient democracy, because the government changes way to often, and it was impossible to do long term policies. So 1945 is also an important date, with a new republic (ie constiution), and universal suffrage.
Public officials in the legislative and executive branches are either elected by the citizens (directly or indirectly) or appointed by elected officials.
In 1870, France was not completly a democracy, because universal sufferage wasn't there.
More accurately, it is a "flawed democracy". France is a representative democracy.
We are a representative REPUBLIC, not a democracy. The last great democracy was in France and ended with the revolution of 1788-89 ( " let them eat cake" ).