"We Believe that men will be punished for their own sins, and not for Adam's transgression" (Article of Faith: 2)
First off, Adam's transgression is the very act of him partaking of the fruit of the tree of knowledge of good and evil which they were commanded to not partake of. When this happened Adam fell from an immortal being to a mortal being which made it possible for all of us to come to the earth and to obtain a body.
It is a common misconception that since Adam and Eve fell, all of their descendants would be born innately evil. We actually believe that children are perfect when they are born and are not punished for Adam's transgression but 'are alive in Christ'(Moroni 8). If you don't believe me then do the simple test of looking into the eyes of the nearest infant or toddler to you. Is there any speck of evil there? Of course not. They are completely innocent. Although innocent, these children, as they grow up are subject to temptations which is where the idea of being punished for their own sins comes in.
When we say punished, we mean that we will be held accountable for our own sins. Since this is the case, we need to resist the temptations that are given to us and we need to always follow the Gospel by having faith and continually repenting (or changing) of our sins.