Lamarckian Evolution is a theory named after French Biologist Jean-Baptiste Lamarck that proposes people, animals and plants pass onto their descendents those traits that they develop in their lifetime, and conversely their descendents lose those traits they don't develop or use. Some examples might be a concert musician passing on his aptitude for music, nimble fingers and fine ear, to his children. These are things he developed in his lifetime, hence those traits / gifts were acquired by his offspring. Or the Giraffe has a long neck from stretching to reach high leaves. This theory is currently not favoured, replaced by Darwin 's more popular theory of evolution , which simply states that things (plants and animals) evolve as a result of chance mutations, favoured by the environment. (Natural Selection).
Now Lamarckian Evolution, on the surface, seems like common sense. We see children grow up into the same professions as their parents - (making use of the gifts evolution and their parents have conferred on them). And it's reassuring, knowing that our life's work won't be lost - those skills we develop will provide an edge for our children. And certain behaviours would certainly seem to run in families - certain types of madness, for example (genetic, but they identified it by the behaviour), intelligence, poverty and class, education - all seem to on the surface reflect a view that "the apple doesn't fall far from the tree". We know that watching or exposing yourself to violence desensitize you towards acting in a violent manner. And children of violence often act out violent scenarios. So it's not a bad theory at all, merely superceded by Darwin's.
Of course we've identified many genes that would predispose you to have a gift in a certain direction - explaining things in the sense that we tend to cultivate those abilities and talents we're good at, and disregard those we aren't. Which is not Lamarckian Evolution, as the hereditary gene causes the behaviour, not the behaviour changing the gene.
But there are increasing numbers of studies and scientists that suggest our behaviours, attitudes and beliefs can alter our gene expression.
Link: Science Daily on Gene Expression/Behaviour, http://www.sciencedaily.com/releases/2011/12/111205102713.htm
Which raises the question of whether the altered gene is passed on? In which case, perhaps there's something to it after all....
Further Reading: Technology Review: A Comeback for Lemarckian Evolution
Update: (May 15 - 09) - Another link seemingly supporting the idea of Lemarckian Evolution.