What does emulsify mean?
To emulsify means to bind two things together that don't naturally bind, like oil and vinegar.
In order to make the combination work, you need an emulsifier, a bridge substance. Common emulsifiers include eggs (to bind oil and lemon juice for mayonnaise) and mustard (for vinaigrette).
To emulsify, you'll need a whisk, blender or food processor. When a recipe doesn't call for an emulsifier to bind, say, a vinaigrette, whisk the oil and vinegar into a temporary emulsion just before serving.
Reference: What makes a mustard "Dijon"?
Reference: Homemade mayo: worth it?
Recipe: Shallot-mustard vinaigrette
Recipe: Kalamata Olive Vinaigrette (Kalyn's Kitchen)