A 'broken' sauce happens when the fat and liquid separate, leaving you with a greasy, curdled mess. It's usually a temperature or ratio issue.
The Mustard Bridge
For vinaigrettes, add a teaspoon of Dijon mustard and whisk vigorously. The mustard acts as a natural emulsifier.
The Egg Yolk Rescue
For warm sauces like Hollandaise: place a fresh egg yolk in a clean bowl with a teaspoon of warm water. Very slowly whisk your broken sauce into the new yolk.