The term color neutral (or colour neutral outside of the U.S.) refers to the ability to start a puzzle on any color. Besides total color neutrality, various partial neutrality are possible. For example, on 3x3, being able to start on 2 opposite (but fixed) colors is known as being double color neutral or opposite color neutral.
In 3x3 Speedsolving
Color neutrality has been explored as early as the beginning of speedsolving. Lars Petrus used color neutrality with the Petrus method to reach World Championship 1982. However, as CFOP became the dominant speedcubing method in the late 90s, most top speedcubers started from a fixed color cross to achieve better look-ahead during F2L. Indeed, while opposite color neutrality was explored and advocated by several cubers, primarily Chris Hardwick, as early as 2004, no serious CFOP user before 2007 recommended color neutrality as a viable option in speedsolving. It was first Rowe Hessler's success starting in 2008, then the meteoric rise of Feliks Zemdegs in 2009, that propelled color neutrality to the forefront of 3x3 speedsolving. As of 2010, however, most CFOP cubers remain fixed-color. Cubers universally agree that the easiest way to become color neutral is to practice color neutrality from the start. Most top cubers who were among the top since before 2009 have so far stuck to single-color cross.
For ZZ and Roux
Color neutrality is especially difficult for ZZ users, who rely heavily on a fixed orientation to decide the edge orientation. The same, to a lesser extent, can be said for Roux users, who prefer to at least have fixed side colors.