There is no official definition of Islamism. The term itself is somewhat controversial and is used in slightly different ways by different people. In general, it can be used to refer to the idea that Islam should be the center of life for a Muslim society. That is, people should base their personal lives on Muslim teachings and (very importantly) society should base its laws on those teachings as well. Many people who talk about Islamism also say that it has a tendency to condone violence to bring about its ends. The term generally is used to describe those Muslims who hold a very conservative view of what Islam teaches. Thus, Islamism can be defined (though not all would agree with this) as a sometimes violent ideology that wants to impose a strictly conservative view of Islam on everyone in a given society.
Before September 11, 2001, most people in the United States would not have said that Islamism was important to the US or even to world history. That changed, of course, with the terrorist attacks that happened on that day. The attacks were conducted by Al Qaeda, a very violent Islamist group. Since then, the United States has been committed to trying to combat Islamism, particularly in its more violent and conservative forms. We can say, then, that much of our foreign policy since 9/11/2001 has been a reaction to the rise of Islamism.
For world history more generally, Islamism is currently important because it has helped to change much of the Muslim world. The “Arab Spring” came about in part because of Islamism. Islamism is helping to drive the civil war in Syria. It is the cause of some of the violence and unrest in Iraq and much of the violence and conflict in Afghanistan. It has helped cause deep divisions in societies like those of Somalia and Nigeria. In short, Islamism is important to world history because it has brought conflict and change to much of the Arab world.