Equality is when things are the same (equal) in some particular way. You had to rethink your campaign for equality in the household when you realized this meant doing your own laundry and sharing the remote control.
Equality means "the state of being equal." It's one of the ideals a democratic society, and so the fight to attain different kinds of equality, like racial equality, gender equality, or equality of opportunity between rich and poor, is often associated with progress toward that ideal of everyone being truly equal. Equality doesn't have to be used only for social ideals, though. There can be equality in weight between two barrels of apples, or between two molecules.