Jean Baptiste Lamarck and Charles Darwin were both prominant biologists who pioneered the science of evolution. Both believed in the concept of evolution where animal populations are changing over long periods of time to better adapt to their overall environments. Lamarck published his theory of inheritance in 1801 and believed that individual organisms made incremental changes to themselves to better adapt. These incremental changes were passed on to their offspring and over time these incremental changes added up to substantial changes for the animal population. An example would be short necked giraffes slowly stretching their necks over time to reach food high in trees. Over enough generations of giraffes stretching their necks incrementally and passing this on to their offspring, giraffes eventually developed long necks.
Darwin published his theory of natural selection in 1859. He believed that any changes an animal population experienced over time were not the result of changes that individual animals made to themselves (inheritance) but rather due to natural selection. Variations among members of a species constantly exist and those variations that are beneficial will allow those organisms to flourish and pass them on to their offspring. Those organisms without beneficial traits will die out and not have offspring. In this way the overall population of a species develops over time. So by his view, long necked and short necked giraffes both existed and the short necked giraffes died out while the long necked giraffes flourished.
Ultimately, the main difference between the two is that Lamarck believed that changes in a species started from the individual level while Darwin believed that species changes always stayed at the population level.