What determines whether the maternal or paternal allele is expressed?
Except in the case of X inactivation in females, and genes on the Y chromosome in males, generally speaking both alleles are expressed in cells. 

A good example is the ABO blood type marker system. If the mother is type AA, she will pass on an A allele, which codes for the A antigen. If the father is type BB, he will pass on a B allele, coding for the B antigen. Therefore, any children will be type AB, as the maternal chromosome 9 will express the A form, and the paternal chromosome 9 will express B.