Quick answer: build — pour into the glass and stir (for fizzy drinks), shake — shake with ice in a shaker (for juices and syrups), stir — mix with a bar spoon (for spirit-only drinks), layer — pour in layers by density (for shots).
Build is the simplest technique and the right place to start. Fill the glass to the top with ice, pour the ingredients from strongest to lightest, give it 2-3 turns with a bar spoon. Used for gin & tonics, highballs, spritzes — anything with a carbonated component that must not be shaken.
Shake is needed when the recipe includes juice, syrup, cream or egg white — ingredients of different densities that need to combine and chill hard at the same time. 10-15 seconds of vigorous shaking is enough: any longer and melting ice over-dilutes the drink. No shaker? A jar with a tight lid works.
Stir is for cocktails made only of alcoholic ingredients: Negroni, Manhattan, Martini. Stirring with a spoon in a mixing glass chills without aerating — the texture stays dense and silky, the drink stays clear.
Layer is the technique for layered shots: pour from the densest ingredient (syrup) to the lightest (spirit) over the back of a bar spoon. The rule is sugar content: the sweeter the ingredient, the lower it settles.
Practice: make one cocktail with each technique — a gin & tonic (build), a sour (shake), a negroni (stir) and a layered shot (layer). Every ingredient is in the BrandBar catalog.