To democratize is to make an organization more democratic, supporting a system in which all citizens have a voice or a vote. When you democratize your school French club, you allow every member to vote on each decision.
You'll most often see the verb democratize in the context of world politics, when countries that have historically not been democracies change their governments to be more democratic. In many cases, other countries, such as the US, try to influence such governments to make this change: they democratize them, in other words. The word democratize comes from the Greek root demokratia, "popular government," from demos, "common people," and kratos, "rule."