Lamarckian inheritance is a synonym for the idea of the inheritance of acquired characters.
As an organism develops, it acquires many individual characters, due to its particular history of accidents, diseases and muscular exercises. Jean-Baptiste Lamarck gave his name to the idea that a species could be transformed if these individually acquired modifications were inherited by the individual's offspring, and further modifications were added through time.
For example, in his famous discussion of the giraffe's neck, he argued that ancestral giraffes had stretched to reach leaves higher up trees. This exertion caused their necks to grow slightly longer and their longer necks were inherited by their offspring, who thus started life with a propensity to grow even longer necks than their parents. After many generations of neck stretching, the result was what we can now see. Lamarck described the process as being driven by the 'striving' of the giraffe. His theory has often been caricatured as suggesting that evolution happens by the will of the organism. However, the theory only requires some flexibility in individual development and the inheritance of acquired characters.
It is actually historically misleading to ascribe the idea of inheritance of acquired characters to Lamarck: it is an ancient idea which dates back as far as Plato.
Can Lamarckian inheritance account for adaptation?