A Moderate Republican is someone who rejects some conservative positions, most notably on social issues such as abortion and homosexuality. Moderate Republicans often support each other rather than conservative candidates and typically reject conservative positions in the Republican Party platform.
Four examples of moderate Republicans are former Senator Olympia Snowe of Maine and sitting Senators Susan Collins, also of Maine, Lisa Murkowski of Alaska, and the retiring Bob Corker of Tennessee. Some might also classify John McCain as a moderate Republican though he has had a largely conservative voting record.
In some states, the rift between moderate and conservative Republicans has become such that Republican primaries almost overshadow the general elections in importance. An example is Kansas, which is a very conservative state that is dominated by the state Republican Party that has factionalized into moderate and conservative wings. The same was even true in Mississippi in the 2014 Senate election there.
Republicans who support traditional conservative positions on social issues but not on economic issues are not typically described as moderate Republicans. For example, in the 2000 Republican presidential primaries Gary Bauer called for an increase in the minimum wage, opposed Social Security reform, and called for curtailed free trade, especially with China. Yet, he was not considered a moderate Republican.