I read somewhere that the two different alleles which we (and other species) inherit from our parents are not expressed equally in our cells. Which genes show allele-specific expression? How and why?
There are a few well known examples of allelic exclusion (which is what you've described):
In females (human and probably other mammals too) one of the two X-chromosomes is silenced (shut down) during embryo development. This means that some cells have one allele (from maternal X) active, while other cells have the other one (from paternal X) active. The actual mechanism behind this is quite complex, but ultimately results in the complete epigenetically-controlled prevention of gene expression from one X-chromosome. This is necessary because otherwise females would produce twice the amount of all proteins on the X-chromosome compared to males.
Another well known example relates to the genes that lead to the building of the B-cell receptors and antibodies. During maturation of the B-cells different segments of these genes are rearranged in a process call VDJ recombination. In order to make sure that the cells produce only one kind of B-cell receptor/antibody one of the two alleles for these genes is silenced after a certain rearrangement step has taken place.