Playing God refers to someone supposedly taking on the role of God for other purposes, also referred to as apotheosis. Alleged acts of playing God may include, for example, deciding who is to live or die in a situation where not everyone can be saved. Usually the expression is used to invoke a precautionary principle or to suggest that someone should refrain from a controversial action.
Philip Ball wrote an article saying that "playing God" is a meaningless and dangerous cliché, in the sense that it has no basis in theology. Instead it was adopted by theocons from the 1931 film version of Frankenstein, and has been used by journalists to refer to things with which they do not agree. The idea is further discussed in Practical Ethics, a blog linked to Oxford University.