When I was a small child, our whole family (eleven kids and parents) were in the local newspaper because my parents were so widely known for their hobby of doing genealogy. We often spent evenings working on family history, writing our own, reading journals from grandparents, or memorizing names and dates of ancestors. My parents even named ten of the eleven of us after ancestors. I'm named after Mette Marie Hansdatter/Hansen, who emigrated to Utah after being converted by missionaries in the late 1800s in Denmark. At home, I learned stories about ancestors who had been pioneers in the early days of the Mormon church, crossing the plains pushing a handcart to get to "Zion," aka Utah. We were very proud of the sacrifices our ancestors had made and this made us determined to "honor" them by continuing to be good Mormons. In Mormonism, family history is basically a requirement for getting to heaven, where you will meet all your ancestors and be part of one giant, eternally bound family.
No, Mormons don't worship or pray to ancestors. No, we don't believe we have lived previous lives. No, we don't believe that by doing temple work for the dead we force them to become Mormon. No, we don't believe that everyone has to or will convert to Mormonism in the after-life.
But yes, Mormons believe that families are eternally sealed to each other in heaven, which means not only our nuclear families, but generations in the past and descendants in the future. Mormons also believe that certain rites are necessary for entry into heaven and that the dead wait for this work to be done so that they can pass from spirit prison to spirit paradise, both of which are places souls wait to be resurrected after this life. Because Mormons believe in a God who is all-loving, we believe that He intends for everyone to have the chance to have these rites done. While we do not believe that everyone will accept these rites and become "Mormon," we do believe it is our responsibility to act as proxy for the dead, in particular our own ancestors, so that they can pass into heaven if they choose to accept these ordinances.
I've heard countless stories in church about Mormons being "prompted" to look in a particular place for family history information, attics, drawers, even phone numbers of certain lost family members who have records. These are inspiring stories about how important temple work and genealogy are to those in the "spirit world," and to God. Mormons will sometimes spend their vacation time going to churches where family members were born so that they can experience a connection with their ancestors, or so that they can find out more information. They will spend time regularly working on genealogy, and feel that it is important to keep their own records in journal form, of photographs, and of important family documents for their own children and later descendants.
When I was a child, my parents would frequently make the cross-country drive from New Jersey to Utah in the summer to visit family. Along the way, my father would stop at cemeteries and hand out notepads and pencils, then send all eleven of us out to write down information from gravestones so that it could be collected and added to the enormous collection of genealogical information which the church stored at the genealogical library in Salt Lake City. When I was a teenager, my mother would take me with her to The Family History Library to record information from microfiche that had come from local parish archives in Denmark. I was instructed to write down everything I could make out, names and dates and family information. Sometimes it was difficult to make out letters or numbers in a strange hand written centuries ago in another language.
Of course, in the age of the internet, this genealogical store of information is a lot bigger and a lot more accessible to everyone, not just Mormons. Anyone can find out information about their ancestors. Anyone can also find the holes in their family trees and seek out more information. Mormon teens are now encouraged to do "indexing," which is seeking out those holes and trying to fill them, and eventually getting temple work done for these family members. Frequently, wards will have a designated "Family History" person who is called and dedicated specifically to helping other ward members with their genealogical work. But if there isn't a specific calling, be sure that in many areas, there will already be a member who is an expert. Mormon missionaries often also try to make connections with non-members through offers to help with genealogy (though if you aren't interested in learning about the church, they will still help you and you can use church resources for free).
As I consider the genealogical work of the Mormon church from a non-spiritual perspective, I think it still has great value. Family history is just another word for history, the history of those who lived so-called ordinary lives and did the working and living and loving and dying in the past that the world may never have noticed before. Family history is also an important way of making sure that people of all nationalities and races are remembered. The Mormon Church combined volunteer efforts of hundreds of thousands of hours to complete the Freedmen's Bureau Project of two million records of freed slaves.
The Pulitzer Prize winning historian, Laurel Thatcher Ulrich, who is Mormon, has done a lot of work in uncovering the "ordinary" life of women in early America through their journals (including The Midwife's Tale) and this is in part because of Mormon emphasis on family history work that focuses us less on the so-called big events of history and more on the way that people have lived in the past. I believe that regardless of your perspective on Mormon temple work, Mormon genealogical work is valuable and honorable and if you haven't done it before, I urge you to go look up your folks and see what you find out about them.