There is no singular, unifying history of Islam. Like most religions, there are differences in beliefs and practices between sects, cities, and scholars. Christianity includes the disparate doctrines of the Catholics, Mormons, Jehovah’s Witnesses, Baptists, and many, many others. They have subtly different origin stories, very different practices, and their own concepts of the “Godhead,” depending on whether they are trinitarian or nontrinitarian.
Islam’s major denominations are Sunni and Shi’a, and each has several branches. Unlike Christianity, there are fewer differences between Sunni and Shi’a Muslims than, say, Anglicans and Pentecostals, so we’ll look at the most general tenets of the two. These general tenets are that the Qur’an is the word of God, Muhammad is God's messenger, God is all-powerful but humans have free will, and everyone will be judged based on their deeds in life. The most important practices that support Islamic beliefs are called the Five Pillars of Islam. These are the shahadah (a declaration of faith), salat (five daily prayers), zakah (alms for the poor), sawn (ritual fasting during the month of Ramadan), and hajj (pilgrimage to Mecca at least once in a lifetime).
The Qur’an is Islam’s principal revelatory text, akin to the Torah or the Old Testament. It dates to the seventh century of the Common Era, when Muslims believe the Prophet Muhammad received it from the archangel Jibril (Gabriel). The Qur’an asserts the supremacy of God, known as Allah, as an all-powerful monotheistic deity. Muhammad is the Seal of the Prophets, God’s final and most important messenger. Unlike Jesus in Christianity, Muhammad is not considered divine.
Three other religious texts central to Muslim belief are the Zabur (Book of Psalms), the Tawrat (Torah), and the Injil (Gospel/New Testament). These books are not considered revelatory due to suspected revisions and corruptions of their original messages. If you’ve recognized the names of some of these books, it’s because Islam and other Abrahamic religions share a great deal of history. Muslims believe that Moses and Jesus were messengers of God, Allah, who spread the gospel and performed miracles on Earth. Although some of their biographical details differ, Christian and Islamic doctrine overlap in many places, including Jesus’s immaculate conception.
The hadith is a collection of recorded religious laws and practices that the Prophet Muhammad lived by. The hadith was written by Muhammad’s followers over several generations and is an important, though not revelatory, Islamic text. Many modern concepts of appropriate behavior for Muslims are derived from the hadith. As is often the case in religion, how the text is interpreted is where disagreements arise. Some groups regard the hadith as an enduring example that should be followed, including its religious laws, called Sharia, that are often severe in comparison to modern concepts of justice. Some follow only the virtues and principles that have to do with morality. Other Abrahamic religions, like Christianity, experience the same divisions. Some Christians believe the bible is literal, while others see it as something that informs, but does not dictate, modern lifestyles. The hadith, for instance, contains several references to a Golden Rule-like principle of compassionate reciprocity, such as:
The most righteous person is the one who consents for other people what he consents for himself, and who dislikes for them what he dislikes for himself.
A Bedouin came to the prophet, grabbed the stirrup of his camel and said: O the messenger of God! Teach me something to go to heaven with it. Prophet said: As you would have people do to you, do to them; and what you dislike to be done to you, don’t do to them. Now let the stirrup go!
There are two major denominations of Islam, Sunni and Shi’a. 90% of Muslims are Sunni. The largest populations of Shi’a live in Iran, Iraq, and Bahrain, but both groups live throughout the world. The differences between Sunni and Shi’a Islam began as a matter of succession following the Prophet Muhammad’s death in 632 CE and centered around the question of whether the new leader should be democratically elected or a family member. Sunnis supported the election of Abu Bakr as Muhammad’s successor. Shi'a supported Muhammad’s son-in-law, Ali ibn Abi Talib. When Ali was murdered, they incorporated beliefs in martyrdom into their belief system. The two sects differ in their concept of succession, and therefore politics, but they share most of the same religious beliefs. Because of this, many people do not identify as either Sunni or Shi’a, but simply as Muslims.
Islam shares much of its concept of salvation and judgment with the other Abrahamic religions. People who have died will be resurrected on the Day of Judgment, and their actions in mortal life will dictate where they go eternally. Those who have lived according to God's will be rewarded with paradise, while those who have lived immorally will be punished in hell.
Regarding your final question, about how to witness to Muslims, I’d recommend most of all a respectful attitude and willingness to listen to views you may disagree with. In general, to “witness” refers to proselytizing by evangelical Protestant denominations. There are many well-meaning Christians who witness by talking about their own beliefs and practices with a receptive, consenting audience. Unfortunately, there are also groups who vilify other religions, referring to them as “cults” and denying their practitioners’ rights to self-determination and religious freedom. For many people, religion is closely aligned with identity. No one appreciates attempted conversion by religious groups who disparage their identity. It’s important to consider how you would feel if someone tried to discredit your beliefs and to treat others the way you’d like to be treated. Learning about Islamic history is a great way to get a better understanding of world religions. To go further, think about reaching out to a member of the Islamic community in your area—maybe even interviewing them. We all benefit when we learn more about each other.