A mixture “breaks” when an emulsion becomes unstable. Butter is an emulsion of fat and water, and when it is creamed with sugar, it incorporates the sugar and air into the emulsion. If other liquids are added too quickly or are too cold or too warm when added, the emulsion will become unstable and “break.”
A broken mixture will look curdled and separated. This does not mean that it cannot be baked. But a cookie or cake baked from a broken batter will have a denser crumb and drier texture.