Forrest Gump is the story of an incredibly kind and gentle person who is also what some people might call "mildly retarded." It's true that he is not too smart, but he is very fortunate, because he has a mother and friend who love him dearly. Forrest is born and raised in rural Alabama, in the Southern United States. He grows up with his mother, who rents out rooms in the family house to people traveling through the area.
Despite his lack of sophistication, and the fact that he was raised far from any major cities, Forrest manages to become personally involved in most of the critical events that take place in American History from the late 1950s until the early 1980s. This includes the Vietnam War, the Watergate Scandal, the Civil Rights and Anti-Vietnam War protest movements, and the Computer Revolution. Forrest even meets three American Presidents as well as Elvis Presley and John Lennon. In a way, this movie is a look at a period of American history through the eyes of a gentle soul who lacks cynicism, but simply accepts things for what they are.
Ultimately, it is about the relationships that Forrest develops through his life. With his mother, who will do anything for him, with his two best friends from his days in the army, Bubba and Lieutenant Dan, and most of all, with Jenny, his true childhood sweetheart. Jenny experiences the changes in American culture from a totally different perspective than Forrest, eventually joining the various protest movements and subcultures of the 1960s and 70s. But in the end, she is faithful to her childhood friend, whose sincerity, kindness and loyalty she would find in no other person.
A special note on language usage: Since much of the movie is narrated by Forrest, you should be careful to not internalize the ungrammatical aspects of his personal way of speaking. Aside from the obvious use of "double-negatives," Forrest also tends to conjugate the past tense of "to be"...