Lamarck was something of a vitalist, in that he believed there was a "life force" that separated living creatures from non-living things. (He believed this had something to do with electricity, the study of which, and its effects on life-forms, was all the rage at the time. See also Galvani's frog experiments, and Shelley's Frankenstein.)

For Lamarckian evolution, the transmissible characteristics were stimulated by, or starved of, this vital energy. So cutting off limbs or tails of animals would not possibly have anything to do with changing heredity. What needed to happen is that a creature needed to actively stimulate the vital energies in that part of their body, and that would cause their offspring to similarly have stimulated vital energies in the same area. So exercise and use would work; so, in turn, would active dis-use. But amputation would probably not.

The broader theory that Lamarck used this mechanism to explain is totally omitted from how biology classes discuss him. It was not Darwinian common descent at all, but a totally different schema. It instead said that, all the time, there was spontaneous generation of simple life forms (this was related, again, to electricity). These simple life forms would then be molded by (Lamarckian) evolution and the "creativity of nature" (a particularly Lamarckian concept) into specific "pathways." So you might end up with a species that goes through an "ape-like" pathway, and becomes an orangutang. Or another species (a different species!) could go through the same pathway and become a chimpanzee. So each species you see is actually a totally different line of descent through time leading to what it is today. It is not a theory of common descent — he would argue that chimpanzees and orangutangs have different branches of descent that do not cross over, except that they went through a similar pathway and thus ended up being very similar to one another. Extinction was not a possibility for Lamarck; a given "line" could change, but never be totally eliminated.

Which is even weirder than the inheritance of acquired characteristics, in my mind! But it reflects how different the evolutionary and biological thinking of the late Enlightenment is from our current one. Lamarckian evolution was not just Darwinian evolution with a different mechanism, it was an entirely different idea about how the animal world was related. (In Darwin's "pangenesis" idea, which appeared in his later writing, hereditary particles called "gemmules" were thrown off by various organs in the body, and more would be thrown off with use. They would then concentrate in the gonads and make up the sperm and eggs, and make a sort of pattern of the new organism when combined with the gemmules of another organism. Darwin's main approach to thinking about organisms was not "heredity" but, like most of his time, "development" — how an egg turns into a full being — which is one of the many reasons this aspect of his theory looks so out-dated to us. Galton and others would, almost within Darwin's late lifetime, bring in the "hard" heredity interpretation that we have come to associate with Darwinian evolution.)

Today we focus on Lamarck's hereditary views, even though that was really only part of his overall argument. Lamarck's views were mostly rejected by contemporary scientists, not so much because of the issue with his ideas about heredity (which were fairly common ones for the time, and the appeal of them is easy to understand — the children of athletes tend to be fairly fit themselves, for example, though we would explain that differently today), but because of his broader evolutionary argument. In particular, the fact that mummified remains of animals did not seem to show any change from present-day animals; that no change was evident despite several thousands of years going by, which was considered to be on par with the age of the Earth at the time. Even those who believed that Genesis was not a literal account of creation did not imagine a billion years of life on the planet, even in Darwin's time. They also argued more generally against his materialism, which was associated (outside of France) with atheism, republicanism, and other -isms that were seen as dangerous and repugnant. Still, Lamarck was read with interest on the margins of things (including by Darwin, when he was a young man), and ended up influencing a lot of popular thinking about evolution before Darwin (such as the very popular Vestiges of the History of Natural Creation, the most popular book on evolution in Victorian England even in Darwin's time).

It is of note that Darwin himself believed in essentially a modified form of Lamarckian heredity — the idea was hard to dispense of! Modern "hard" heredity (no acquired characteristics) did not really get fully developed until very late in Darwin's life (in part by his cousin, Francis Galton, who tried to test Darwin's neo-Lamarckian "pangenesis" concept), and did not really flourish until the 20th century.

In biology classes, Lamarckian evolution is used primarily as a foil to Darwinian evolution; its invocation is not really about understanding the history of evolutionary thinking, but in showing that a) there are other (wrong) evolutionary theories possible beyond Darwinism, and b) illustrating natural selection by contrast. Actually understanding Lamarck's work in context requires understanding a lot of Enlightenment biological concepts — really working to understand it as it was in its day, not just picking out one part of it. As a historian of science I think such things are useful and interesting to know, but I also understand why biology teachers don't get into the details, or even portray the history all that accurately. (This is not limited to biology class; almost all of the history of science you get in science classes follows a similar pattern of error and misunderstanding, deployed for pedagogical effect.)

The biggest error, from the perspective of teaching students about science, is the model of science itself that comes up in such a context. The idea that "person 1 put out theory 1, and then an experiment showed that theory 1 was wrong, and person 2 suggested theory 2, which the evidence showed was right, and that's why we believe in theory 2," makes for a nice, neat fairy story about how science works, but it is not actually how things worked historically.

The best overall book on this and everything else you could put under this heading is Peter Bowler's Evolution: The History of an Idea, which handles Lamarck very well.