I see many Muslims around the world differentiate themselves from Islamists. What is the difference between an Islamist and a Muslim?
Islam is a religion: a set of beliefs (and associated practices) and traditions.
A Muslim is someone who believes the tenets of Islam and follows its practices and rules.
An Islamist is someone who believes that the rules of Islam should be enforced on non-Muslims or should in some way inform and influence state legislation.
Note the important distinctions here: to call someone a Muslim is to describe their religious beliefs and practices, while to call someone an Islamist is to describe their political beliefs and practices. (It is even theoretically possible, though vanishingly unlikely, for someone to be an Islamist without being a Muslim. A person might not believe the tenets of Islam, but nonetheless think its laws are a good idea and should be enforced on everyone.)
There is also be a wide spectrum of beliefs within Islamism. Some may believe that full Islamic law should be enforced on everyone, while others may think that secular law should be to a certain extent influenced or informed by Islamic practice.
Also note that the word Islamist is used most often to describe terrorists or other people who use violence to enforce their goals. This is not actually part of the definition of the word — it’s perfectly possible to be a peaceful Islamist who attempts to influence legislation in an Islamic direction through persuasion or at the ballot box — but it’s a common association and may cause many Muslims to clearly disclaim the position.
A similar distinction in terminology exists between Christians and Christianists, by the way. The American religious right, for example, are Christianists, most of them non-violent ones.