The statements below are all according to my understanding of Mormon theology.
Spirit matter is a kind of matter that is "more refined" than normal matter. Everyone on Earth was originally an intelligence uncreated, co-eternal with God. God and one of his wives became our spiritual parents when they gave spirit bodies to our intelligences. God has a body of flesh and bones (but not blood because blood is bad for some reason). To Mormons the term "Father in Heaven" means the father of our spirits.
Jesus Christ was the first intelligence given a spirit body so Christ is God the Father's firstborn in spirit, but also firstborn in Flesh (via his sexual union with the human Mary). We are all spirit children of Heavenly Father. Satan was also a spirit child of God, a "son of the morning" (because he was one of the earliest?), who rebelled against God. Part of his punishment is never receiving a physical body.
When Christ told Peter "Flesh and Blood hath not revealed this [testimony] unto you, but my Father which is in heaven", it meant a human being with blood didn't reveal it to Peter but God who has flesh and bones (but no blood). Spirit fills his veins.
The Earth was created so that our spirit bodies could eventually be merged with a physical body through the process of resurrection so we could be like our heavenly parents. When we are born (some time between conception and birth) our indestructible spirit + intelligence enters the fetus and remains in the body until the person dies (or has a temporary out of the body experience, such as some kinds of visions). The physical body + spirit body + intelligence = the soul. Much of our personality is not genetic, but is due to the personality of our spirits (what we were like before being born).
At death the spirit+intelligence leaves the physical body and enters a temporary state in the spirit world. Eventually the spirit+intelligence is joined with an indestructible physical body which is called "immortality".
"Eternal life" is not life for ever, but life like God's, since "Eternal" is one of God's many names. Eternal life is existing in a glorious realm on a sphere of glass with many wives with which worthy men can presumably have sex to their heart's content. Women have the glorious opportunity to share the men with their other wives giving spirit bodies to other intelligences out there waiting for a chance to progress like we did. We'll create Earths and be Gods ourselves. Cool, huh? 10%+ of your Earthly income is one of the few requirements to eventually reach such an exalted state.
Some LDS have suggested that we don't hear much about Mother in Heaven because it could cause conflicts as people argue about who's mother is the best. Others have suggested that her role was primarily in our pre-existence (before coming to Earth) sort of like the mother's role in child birth and rearing on Earth.
The Earth has a spirit body inside its physical form. It was born (created), baptized (in Noah's Flood) and will be burnt after the 1000-year reign of Jesus Christ on Earth. Then the Earth will be converted into a huge Urim and Thummim, a sphere of glass, on which those who are exalted = those who have Eternal life will dwell.
Animals also have spirits, but not advanced, God-like spirits such as our own. I don't know if plants are supposed to have spirits.
Mormons believe in resurrection, not reincarnation.