Jean Baptiste Lamarck (1744-1829) was first trained as a military officer. However, after a fall on his head, he quit that career to take up his real passion: botany. His first work, on the flora of France, appeared in 1778, and he was invited to Paris by the leading natrualist, Georges Buffon, to work at the Royal Garden of Plants. Following the French revolution, the Royal Garden was reorganized as the Museum of Natural History, and professorships opened in a wide variety of fields. Lamarck applied for the prestigious professorship in vertebrate zoology, but was not successful; instead, he was offered the less prestigious professorship of the invertebrates -- containing the catch all category of "worms" that Linnaeus had included in his earlier classification.
Lamarck eventually separated out no less than 10 sub-divisions within the invertebrates, ranging from the Infusorians (with the simplest nervous system and internal organs) through insects to the molluscs (the most complex of the invertebrates -- e.g. shrimps, crabs, and lobsters). Within the vertebrates, he distinguished four classes - the fish, reptiles, birds, and mammals (as in Linnaeus); to which he added, between the reptiles and birds, the class of "monotremes" to include the newly discovered "duck-billed platypus" from Australia (today included as a mammal) -- which combined the bird-like habit of laying eggs, in an animal which provided milk to its young. This system was set out in the multi-volumed History of Animals without Backbones published from 1815-1822.
Although most respected among his colleagues for that work, our interest will focus on an earlier, more speculative work: the Philosophical Zoology of 1809. Here, for the first time in biology (the term "biology" was coined by Lamarck to include zoology and botany) a theory of species change was adopted as a central concept. What Lamarck did was to take the classes he had defined in his systematics and reconceptualize them as a sequence of transformations of some into others over time. The following diagram from his Zoological Philosophy illustrates this concept:
For our purposes the main thing to note is the trail of downward lines, indicating descent from earlier, simpler classes to later, more complex ones. The details of the various classes are secondary, as are the details of how this affects each species within a class. Note at the bottom of the descending lines the following mammalian classes:
Lamarck developed a theory which identified three factors to account for this evolutionary process:
Taken together, Lamarck's three factors constitute his theory of evolution. You can read his own description of the theory here, beginning with a discussion of the evolution of the body of the seal. But taken individually, each of his factors is false. Lamarck was a creative thinker who deserves a major place in the history of biology. He challenged the theory of species fixity; but he could not complete the scientific revolution begun by Vesalius, Harvey, and Linnaeus. This was due to the excessively speculative nature of his theory, especially the "perfecting factor", which does not provide an explanation meeting the canons of scientific discourse: he had no physical mechanism, only a metaphysical claim. Darwin will attempt to change that with his theory of evolution chiefly, though not solely, via "natural selection".