What is the difference between wild, ancestral and risk allele? How is that related to variant, minor(ity) or major(ity) allele?

Wild allele typically references the majority allele in a specific population although the term major allele is often used too. Major allele commonly references the most common allele across all populations (but not always, it may only be referencing a particular population as mentioned before!). Minor allele is the corresponding minority or least common allele in a specific population or across all populations and is also sometimes referred to as variant. It is important to realize major and minor alleles can vary dramatically across populations in their frequency. What is major for a ACE gene mutation in Caucasians, may be the minor allele in Latinos for example.
Ancestral allele refers to the allele found in the chimpanzee: the last common ancestor of humans and great apes; many times major, wild and ancestral allele are the same across all ethnicities. It has been Stratagene's policy, when the major and minor allele vary by population, to use the ancestral allele as major allele reference point. Risk allele, aka effect allele, gets used different ways in research, but typically it refers to an allele which may be protective or harmful.