A Mormon ward (local congregation) is run by a lay ministry rather than paid clergy, so there are different members of the ward speaking each week. Usually there are 2-4 speakers, and the addresses that they give are called "talks" (Mormon Terminology, Part 1). How are those speakers chosen?
Once the speaker accepts the request to give a talk, he should start to think about his topic and research it. He should also make sure that he prays that God will guide him in his preparation. The given topic is often very broad, so it helps for the speaker to consider what life experiences he has had that have to do with the topic. Choosing personal experiences to share in the talk helps to narrow down the topic a little, and it might just be why God chose that person to speak on the given topic. Someone, or several people in the ward, might need to hear about what the speaker learned from a life experience. Many Mormons will tell you that they have heard talks before that have completely changed their lives. When the speaker prepares and gives his talk with God's help, it's highly likely that many people who hear the talk will receive significant help for their own lives.
There isn't a prescribed style for giving a talk, but most people start by introducing themselves, and as they get into their topic, they try to teach doctrine using the scriptures, quotes from General Authorities (well-known church leaders), and personal experiences. Each talk should last about 5-20 minutes, depending on how many speakers there are in the program and whether there is an intermediate musical number or not. The speaker will often end with his testimony and close in the name of Jesus Christ. Mormons typically close in Jesus' name as a way to affirm that in all they have said, they have tried to convey a message that the Lord would approve of.
Using a lay ministry to supply speakers has its advantages and disadvantages. Speakers will make mistakes every now and then, so you might hear some false doctrine. But Mormons believe that we shouldn't automatically take everything speakers say at face value anyway, so you want to make sure that as you're listening, you pay attention to whether the Spirit is confirming to your heart or not that what the speaker is saying is true. Another disadvantage is that you might find some speakers' styles very tedious, over-your-head, or otherwise hard to follow. However, it's a new set of speakers each week, and some people are very hilarious and entertaining. Other people have inspiring stories or musical voices that you could spend all day listening to. Probably one of the most beneficial aspects of having different speakers each Sunday is the variety of personal experiences that are shared. When one speaker tells about how God helped him to overcome an addiction, or when another tells of how she came to know that God was actually real, the congregation have yet another reason to believe that their own testimonies about God are legitimate.
The system also serves to benefit the speakers themselves. They will probably be the ones to learn the most from their topic. Speakers also get the chance to practice leadership and public speaking skills in a supportive environment. Speakers might also feel compelled to draw closer to God as they seek to know what God wants them to say. When it becomes your turn to give a talk, you can't deny that it would be nice to have some divine inspiration to help you out. When people start drawing closer to God, lives start changing in wonderful ways.
If you would like to read more about how the church uses a lay ministry, see this related article on Mormon.org: Why don't Mormons have paid clergy?