French naturalist Jean-Baptiste Lamarck would arguably be one of the people who had the strongest influence on Darwin's theory of natural selection. As quoted in our textbook, no one "attempted to explain the evolutionary process"- until Lamarck came around. Lamarck came up with the 'inheritance of acquired characteristics' theory- essentially, that a trait acquired by an animal during its lifespan can be passed on to its offspring. Though later to be proved wrong years down the road, it was one of the first scientific attempts to explain how evolution worked.
Darwin's theory of natural selection basically states that animals with 'favorable' traits for their environment would be able to live longer and reproduce, whereas those with 'undesirable' traits would not. This would lead to the creation of a new species.
"In order for traits to evolve and change, they MUST be heritable. This means they must be passed
on reliably from generation to generation through reproduction. This was actually a problem for
Darwin as he did not know how traits were passed on. The current thinking of that time argued for
Fluid or Blending Inheritance, which actually blends out traits". That bullet point under "How evolution works" is the statement that I feel is most directly affected by Lamarck's theories.
Without the influence of Lamarck's theories, Darwin would not have been able to realize and proves his theories at the time that he did. Lamarck was able to give Darwin a sort of 'crutch' to start his explanations with. Without Lamarck, Darwin could have easily still produced his theory of natural selection, but it would have taken him much longer to do so because he first would have had to try and explain how traits could be passed down before he realized in his theories that favorable traits would live longer in their habitat.
During the time of Darwin's publications, religion was still a very huge role in the lives of many people, and most people were not too keen on scientific explanations of the world came about. He thought that his works would be viewed as a serious threat to the religion of the time, and indeed it was. Many of his friends and peers were concerned about with the 'threats to the status quo'.
Link for more information about Lamarck- http://www.ucmp.berkeley.edu/history/lamarck.html