Family History Work Within The Mormon Church.

The Church of Jesus Christ of Latter-Day Saints has participated in family history work and genealogy for over 150 years now. My personal knowledge of it as a member of the LDS Church is that within the temple in Kirtland Ohio, Joseph Smith, who's known to be the first prophet of the LDS Church and the founder, along with Oliver Cowdery, who was a scribe for Joseph Smith during the translation of what is called The Book of Mormon, had the Prophet Elijah descend down upon them declaring great and marvelous things. This revelation, resulted in the Church getting involved in a strong amount of family history work. The Prophet Elijah taught them about the works and blessings that come from family history work. It is our belief that the Priesthood Keys were restored upon the earth during this revelation, witch now makes it possible to be able to extend a helping hand to our deceased ancestors and redeem them. This is done within the LDS Temples along with a lot of other saving ordinances.

                     Joseph and Oliver in the Kirtland temple with Elijah                

LDS Church members have submitted millions of names of ancestors to have baptisms done on their behalf, but its not just baptisms.
There are are several ordinances that can be bestowed upon members within the Church. Baptism, Confirmation, Initiatory, Endowment, Sealing to Parents, and Sealing to Spouse.
 I'm not sure weather you know the last three that I mentioned or not so I'm going to explain what it is below.

•  Endowment is a gift given to members who are worthy to enter the LDS Temples and are of the required. You receive garments witch is believed to protect you from temptation and evil when you wear it. 
• The Gift, Sealing to Parents, witch is also bestowed upon members in the Temple,  is an ordinance given to those that if they're and their parents are willing to remain faithful and obedient to our Savior, you'll remain with your family after death for eternity. This sealing power is also done for married couples within the Church, witch is usually done during marriage. Hence Sealing to Spouse

All of these blessings are done for the dead. Even though they're done for our ancestors, it does not mean that our ancestors will accept it. Its their choice to act on it. otherwise the blessing is taken away from them and they continue as they are.
After Jesus died on the Cross, our belief is that he visited the Spirit World and taught the people who are dwellers within. Moses, Adam, all the prophets and people in the Old Testament, and also the prophets and followers in the first 300 pages in what we call The Book of Mormon got to see our Savior in the Spirit World. Jesus then returned to his body and resurrected after his visit.
We believe that all those upon the world who die, ether go to the Spirit World or Spiritual Prison. The Spirit World is filled with the people who made the more righteous decisions in life, while the people in Spiritual Prison made the more wicked decisions then the better ones in life. It is among those in the Spirit World and Spiritual Prison, people that have not had the opportunities to choose to have the true knowledge of the gospel in their life. We believe that their is still missionary work being done in the Spirit World. The ones who receive the knowledge of the gospel and desire to receive spiritual ordinances are the reason why we perform special ordinances within our temples on their behalf. That way they can have these blessing.
This is why we do family history work and genealogy, that way our ancestors can also receive happiness.

If you have any questions, you can post them on the comments. I'm a family history consultant within the Church and I'll try to answer all the questions you have on this topic. Otherwise, check out the LDS Churches website for family history work. https://familysearch.org/ 

Thank You