My discussion of the doctrine of prophecy in Islam will be taken largely from the Facts On File Encyclopedia of Islam, by Juan Campo, 2009. The entry on prophecy is on pages 559-561.
Prophets and prophecy have a very important role in Islam. The role of the prophet is to provide a conduit between God and human beings. There are two types of prophets in Islam. Let us examine these two.
First, there is the prophet who receives messages from God through angels. (In Arabic, these are called rasul.) The angels deliver the message to the prophet who, in turn, delivers it to the people. Muhammad was a prophet of this sort, though he was not the only one. Islam recognizes many figures from the Jewish and Christian traditions as prophets. These include people like Moses, Solomon, John the Baptist, and Jesus. Muhammad is most important because he was the final prophet of God.
These sorts of prophets tend to have a number of traits in common. They are believed to be chosen by God. Through God, they have special knowledge that is not given to other human beings. They are much more virtuous than other human beings. Finally, they come to warn people of the consequences of their bad actions, but also to give them hope for salvation.
Second, there is a type of prophet that is much more common than the first type. (Called nabi in Arabic.) This type of prophet is more of a law-giver than an evangelist. This prophet understands what sorts of rules God wants to lay out for his people. This prophet then explains those laws to the people. There can be many of these prophets as God uses prophets from different groups of people to speak directly to those people.
In these ways, prophets and prophecy are very important to the Islamic faith.