Partisanship refers to the act of forming political opinions favoring one group over others based on prior allegiance rather than because of the innate value of that group's proposals. A "partisan" statement is one which shows unwarranted favor towards one group, while a "non-partisan" statement is one which treats all sides fairly.
These words are frequently misused, however, to refer to whether or not all sides in a conflict are favored equally regardless of how much sense they make.
Example: when a 2008 study was released showing that the Bush II administration had deliberately and systematically lied in order to convince Congress and the public of the need to invade Iraq, pro-Bush critics spoke in tones of outrage about how clearly "partisan" this was – despite the fact that the accusations were backed up with extensive evidence, which clearly negates the charge of partisanship.
When used in this way, the charge of "partisanship" is a rhetorical tool for manipulating the understanding of naive listeners by moving the apparent "happy medium" away from a sensible viewpoint and towards one which makes less sense by framing the discussion as a choice between two equally valid possibilities (when in fact one of them is much saner than the other), thus inviting the listener to commit the fallacy of moderation. (This is essentially a variant of the Overton window technique.)
Another example: intelligent design is a load of crap not held as valid by any serious scientists (including many who are Christian), but the religious right argues that it is only fair to teach ID alongside evolution in classrooms, to give "both sides of the story"