Mari vs. époux
In French, both mari and époux translate into English as "husband", but they are used in completely different contexts based on the level of formality. The choice between these two words depends mostly on whether the situation is everyday and conversational or highly formal and administrative.
Mari
The word mari is the most common, everyday term used to mean "husband" in standard, casual, and spoken French.
Mon mari cuisine tous les soirs pour la famille.
(My husband cooks every night for the family.)
Elle cherche un cadeau d'anniversaire pour son mari.
(She is looking for a birthday gift for her husband.)
Je vous présente Paul, le mari de ma collègue.
(I introduce you to Paul, my colleague's husband.)
Son mari travaille comme ingénieur dans une grande entreprise internationale.
(Her husband works as an engineer in a large international company.)
Les maris attendaient patiemment dans la voiture.
(The husbands were waiting patiently in the car.)
Tu devrais en parler avec ton mari avant de prendre une décision.
(You should talk about it with your husband before making a decision.)
Époux
The word époux is a formal, administrative, or literary term for "husband" or "spouse", frequently found in wedding ceremonies, legal contracts, and official documents.
Le futur époux doit signer ce registre civil à la mairie.
(The future husband must sign this civil register at the town hall.)
Cet appartement appartient conjointement à madame Dubois et à son époux.
(This apartment belongs jointly to Mrs. Dubois and her husband.)
En cas de décès, l'époux survivant hérite de la totalité des biens.
(In the event of death, the surviving spouse inherits all the assets.)
Veuillez indiquer le nom et la date de naissance de votre époux sur ce formulaire d'assurance.
(Please indicate the name and date of birth of your husband on this insurance form.)
Les jeunes époux ont décidé de passer leur lune de miel en Italie.
(The young spouses decided to spend their honeymoon in Italy.)
La reine marchait dignement à côté de son royal époux.
(The queen walked with dignity next to her royal husband.)
Summary
In short, you should use mari for everyday, casual, and standard interactions when referring to a "husband". You should switch to époux when dealing with legal contexts, administrative paperwork, or highly formal ceremonies where an elevated tone is required.