The word "decline" can mean "decrease," "deteriorate," "move downward," or "politely refuse." Its source, the French verb décliner, can have all of these meanings and more.
Most of these other meanings stem from a more specialized grammatical one. To "decline" a noun, pronoun, or adjective is to list all of its forms according to case, number, and gender. You don't have to worry about doing this in French—it only applies to certain languages, such as Latin and Ancient Greek. But décliner can refer to a similar activity of enumerating, presenting something in various forms, offering a range of something, laying out all its different facets.
Because décliner has such a wide variety of meanings, its translation is highly context-specific. For example, you can use it to talk about a fashion designer "presenting" all the styles of his latest collection on the runway:
Du blanc, du noir, presque exclusivement, tous les codes déclinés inlassablement,
Almost exclusively white and black, all the styles presented tirelessly,
Caption 5, Le Journal - Défilé de mode - Part 2Play Caption
Or you can use it in the sense of "depicting" several aspects of something:
...des travaux de couture d'une jeune femme qui décline un petit peu l'Alsace sur du tissu
...some sewing projects from a young woman who kind of depicts the various faces of Alsace on fabric
Captions 18-19, Alsace 20 - Mangez bien, mangez alsacien!Play Caption
Businesses often use décliner to advertise a product available in various forms. When Lionel visited a madeleine shop in Liverdun, the owner used it to refer to the different flavors she sells:
Nous l'avons déclinée à la mirabelle... -Oui. et à la bergamote.
We've adapted it with mirabelle plum... -Yes. and with bergamot orange.
Captions 32-33, Lionel - La boutique de madeleines de Liverdun - Part 2Play Caption
This restaurant owner in Nice uses décliner in a somewhat particular sense. He's not talking about the different forms of socca he offers, but rather all the times of day people order it:
Ça se décline comme ça, et on peut en manger vraiment à n'importe quelle heure.
It's available like that, and you can really eat it at any time.
Captions 34-35, Le saviez-vous? - La socca, spécialité niçoisePlay Caption
If you see décliner on a form you're filling out, or hear it from an administrative official, you're being asked to provide information about yourself:
Déclinez votre nom et adresse.
State your name and address.
Don't forget that décliner also has all the senses of the English "decline": "decrease," "deteriorate," "move downward," "politely refuse."
We've now "declined" all the meanings of décliner!