Charles Darwin came up with his theory while studying birds in the Galapagos Islands. There would be variation in a species. He hypothesised that if a bird had a beak, they would survive for longer and reproduce more because they were better suited to getting food. The bird would pass on its genes and characteristics to the offspring. He found that over time all of the organisms in that species will have that characteristic, this was evolution.
Charles Darwin's theory was controversial at the time as people believed that God created all. his theory was accepted 50 years later when inheritance and variation was discovered, proving that offspring inherit the genes from their parents.
Lamarck believed that parents give their offspring the characteristics that it had acquired during its lifetime as environmental variation. For example, if a giraffe stretched its neck to reach some leaves, then when it reproduces it would pass on this trait to the offspring.
Why it was rejected
His hypothesis simply didn't work. If a giraffe's tail was cut off, and it reproduced, the offspring would still have tails. His theory was definetly proven wrong with the discovery of DNA - and how stretching your neck does not change the DNA in your gametes.
What is Natural Selection?
Natural Selection is where there is variation in a species. For example, an adaptation such as a long neck. The organisms with the long neck would survive longer and reproduce more as they have more food up high in the trees. The non-adapted ones die out as there is heightened competition and they cannot adapt to the environment.