Who can be saved? The short answer: everyone. The long answer is a bit more involved.
The Church of Jesus Christ of Latter-day Saints teaches that the word “saved” has many meanings. Here’s a brief description of all the ways Mormons use the word “saved”:
- Saved from physical death, or resurrected. Mormons believe that everyone is saved from death, or resurrected. Because of Jesus Christ’s resurrection, everyone who ever lived on the earth will be resurrected. So in that sense, everyone is saved, no matter how righteous or wicked they were on earth. Resurrection is a gift to all people (and all living things) who have ever or will ever live on the earth.
- Saved from sin. Mormons believe that everyone can be saved from sin if they repent and come unto Christ. In the gospel of Jesus Christ, everyone can repent and be forgiven, and thus saved from sin. Unlike resurrection, salvation from sin will only come to us if we do what the Savior has asked. It is still a gift, because it is through the Atonement that we are saved from sin, but we must do our part in order to be saved from sin.
- Saved can mean being born again. Mormons believe that being born again means to be baptized and receive the gift of the Holy Ghost. Again, everyone can be saved by coming to Christ, being baptized, and receiving the gift of the Holy Ghost.
- Saved from ignorance. Mormons believe that salvation from ignorance comes from learning about the gospel of Christ and coming unto Him. Everyone can be saved from ignorance by coming unto Christ. Again, we must choose to be saved and take action.
- Saved from spiritual death. For Mormons, spiritual death means separation from God. Everyone can be saved from spiritual death through their actions and the grace of Jesus Christ. We are not automatically saved from spiritual death; it is conditional on our works in this life.
- Saved can mean exaltation in heaven, or eternal life. While salvation can mean going to heaven, Mormons believe that one type of salvation refers to eternal life, meaning being exalted to the highest kingdom of heaven and then living in the presence of God and Jesus Christ for eternity. This kind of salvation will only happen after we have been resurrected. In this sense, nobody will be saved in this life; however, the things we do in this life can lead to eternal life. Everyone can gain eternal life so long as they make and keep the necessary covenants, or promises with God, in this life. 
So, to summarize: Mormons believe that everyone can be saved. Mormons use the word saved in many contexts. All people who have lived or will live on earth will be saved from physical death; this is a gift from our Savior. In order to gain salvation in every other sense, we must act while on earth. Mormons believe that salvation comes through both works and grace. Salvation from spiritual death comes through both the Savior’s grace and our works while on earth. The other types of salvation require us to do works in this life; however, we ultimately cannot be saved from sin, ignorance, or receive eternal life without the Savior’s grace.
These different ways in which the word salvation can be used seem only to encompass people living on earth now. Many of them are dependent on our work while we are on earth. In order to gain salvation from sin, we must repent and be baptized while on earth. However, millions and millions of Heavenly Father’s children did not have the opportunity to be baptized while on earth. The gospel of Jesus Christ does not leave them out in the cold for eternity. Mormons believe that all of Heavenly Father’s children, even those who have died without knowledge of the gospel or an opportunity to act, can be saved.
The gospel of Jesus Christ is inclusive. Heavenly Father is a loving God, who does not condemn many of His children simply because they lived in a time or place without the gospel or opportunity to act and be saved. Through ordinances for the dead performed in holy temples, our family members and loved ones can be exalted even after death. Children who die young, without baptism, are exalted in heaven, not condemned (see Moroni 8:9-12). All people who desire to be saved from sin and ignorance can be if we choose to be, and perform the works and make the covenants with Heavenly Father.
Salvation, or being saved, is ultimately possibly only through the Atonement of Jesus Christ. It is because of Him that we can be resurrected, forgiven of our sins and transgressions, and eventually return to live with Him again.