According to Islamic tradition, in about 610 AD the angel Gabriel appeared to a man named Muhammad in the city of Mecca and informed him that he was God's final prophet. Muslims (the followers of Islam) believe that over the years that followed, until Muhammad's death, God revealed the Koran to Muhammad. They do not believe that Muhammad founded a new religion because they believe that God was "correcting" the true religion: Muhammad restored the original faith as it was always supposed to be. That being said, Islamic tradition suggests that the teachings of the prophets that preceded Muhammad (including Abraham, Moses, and Jesus) had all been misinterpreted or distorted over time. While these prophets are revered, only Muhammad's words are the accurate, true word of God.
To a Muslim, the term "unbeliever" or "non-believer" refers to anyone who rejects Islam, its principles, and the belief that only Muhammad's words are the accurate, true word of God.
Just as the Bible is interpreted differently by Christians of differing teachings (Roman Catholics and Jehovah's Witnesses, for example, do not share the same beliefs and practices), the Koran is interpreted differently depending on which Muslim you talk to as well. Some claim that the Koran is peaceful and instructs Muslims to "call non-believers to Islam," meaning that Muslims should peacefully evangelize (much like the Jehovah's Witnesses do when they go door to door). Yet other, more militant Muslims believe that if non-believers fail to convert, then they must be fought or even killed. The most extreme of these Muslims are terrorists (a small percentage of all Muslims).
While we're talking about it, maybe it's worth pointing out that many religions have extremists, not just Muslims, and those with differing viewpoints often suffer as a result. You'll often hear Muslim terrorists compared to Christian anti-abortionists. Many Christians believe abortion is murder, but very few Christians commit violence against abortion clinics or the doctors who work in them. Likewise, the overwhelming majority of Muslims aren't terrorists; they don't support the idea of achieving religious gain through violence, and might strive for the same goal through more peaceful means.