Couper vs. hacher
The French verbs couper and hacher both relate to the English concept of cutting or chopping, but they describe different methods and results. While couper is the general term for cutting or chopping something into pieces, hacher specifically refers to chopping something very finely, mincing, or grinding.
Couper
The verb couper is the most common and general word used to express cutting, slicing, or chopping something into regular pieces, parts, or chunks.
Il va couper le pain pour le dîner.
(He is going to cut the bread for dinner.)
Elle doit couper les légumes en gros morceaux.
(She has to chop the vegetables into large pieces.)
Le cuisinier va couper la viande avec un couteau bien aiguisé.
(The cook will cut the meat with a sharp knife.)
Tu peux couper ce gâteau en huit parts.
(You can cut this cake into eight slices.)
Nous allons couper l'arbre qui bloque la route.
(We are going to chop down the tree that is blocking the road.)
Hacher
The verb hacher means to chop something into tiny pieces, to mince, or to grind, often using a rapid motion, a heavy blade, or a specialized machine.
Tu dois hacher l'oignon finement pour la sauce.
(You must chop the onion finely for the sauce.)
Le boucher va hacher le bœuf pour faire des hamburgers.
(The butcher will grind the beef to make hamburgers.)
Il préfère hacher les herbes fraîches à la main.
(He prefers to mince the fresh herbs by hand.)
La machine sert à hacher les branches tombées dans le jardin.
(The machine is used to chop up the fallen branches in the garden.)
Elle va hacher l'ail avant de l'ajouter dans la poêle.
(She will mince the garlic before adding it to the pan.)
Summary
In short, you should use couper for general cutting, slicing, or chopping into standard pieces, and choose hacher when you need to mince, grind, or chop something into very small, fine pieces.