Darwin's theory has been supported by a lot of evidence. Lamarck's Theory of Inheritance of Acquired Characteristics has been disproved. This was done in two major ways. The first is by experiment. We have seen through many real examples and observations that changes that occur in an animal during life are not passed on to the animal's offspring. If a dog's ears are cropped short, its puppies are still born with long ears. If someone exercises every day, runs marathons, eats well, and is generally very healthy, the fitness is not passed on and the person's children still have to work just as hard to get that fit and healthy. These and other examples show that Lamarck's theory does not explain how life formed and became the way it is.
The other way that Lamarck's theory has been proven wrong is the study of genetics. Darwin knew that traits are passed on, but he never understood how they are passed on. During the time when Darwin's first book first came out, Gregor Mendel, who discovered genetics, was just starting his experiments. However, now we know a lot more about genetics, and we know that the only way for traits to be passed on is through genes, and that genes can not be affected by the outside world. The only thing that can be affected is which gene sets there are in a population, and this is determined by which individuals die and which ones live. This is the other way that we have learned that the fruits of an animal's efforts can not be inherited by its offspring.