Oil and vinegar do not mix. The only way to combine them is to whisk them together so strenuously that the vinegar breaks down into tiny droplets. The two fluids are then effectively one homogeneous mixture, called an emulsion.
To enable an emulsion to stay stable, add an ingredient that acts as an emulsifier, in this case mustard. Emulsifiers form a barrier around the droplets in an emulsion, keeping them from recombining and separating out. The mustard might not be enough to affect the flavor of the dressing, but it can have a serious impact on the chemistry of the mixture.