How much is the Lord of the Rings Movie Trilogy like the original Tolkien novels? How Faithful was Jackson to Tolkein?

Different parts of the movies are like and unlike the books, though it depends on which scenes. Some scenes have been completely made up. From memory, I'll give you some examples which shall probably be mostly from the first movie.
The scene at the beginning of the first movie (explaining Sauron and the Last Alliance) wasn't in the book; that information was collected gradually in the book.
The scene where Gandalf arrives in the Shire is different. In the books, he drives his horse and cart to the front door of Bilbo's house. In the films, he is met by Frodo in the woods, and they travel the last few miles together.
Merry and Pippin's role is different. In the films, the literally crash into Frodo and Sam and join their journey. In the novel, they notice he is planning to leave and help him in various ways before joining the journey.
Legolas and Gimli's characters were expanded greatly in the movies. They are secondary characters in the books, with Frodo, Sam, Gandalf and Aragorn taking the main roles. Most of the humorous scenes containing them were created by Peter Jackson. However, the scenes that show them becoming better friends between leaving Rivendell and arriving in Lorien are accurate.
However, the role of Aragorn was cut down in the films, to allow other characters the limelight. The books contain much more about his character, his history and his destiny to be king.
The Council of Elrond was also different in the films: in the books it is both longer and contains less people. They went into the history of the different elves and the ring and the different options about what to do with it.
The role of Arwen – and subsequently the romance between Aragorn and Arwen – was also increased. She played a much smaller role in the books. In the movie, she replaced another elf of Elrond’s household in finding Frodo and defying the Black riders.
Possibly the characters who were portrayed in the movie as closely as they were written are (in my opinion) Eowyn and Gandalf.