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)
