In "Dimanche soir" (Sunday Night), the slam poet Grand Corps Malade declares his love for his wife in beautiful lines such as:
Je l'ai dans la tête comme une mélodie, alors mes envies dansent
I have her in my head like a melody, so my desires dance
Cap. 17, Grand Corps Malade - Dimanche soir
If you didn't see the translation, you might have guessed that envie means "envy." And you would have been right!
Vous ne connaissez que l'envie, la hâte, la rage de les tuer.
You knew only envy, haste, the urge to kill them.
Cap. 60, Il était une fois... L’Espace - 3. La planète verte - Part 6
However, besides désir, envie is also the word for "desire." While un désir is a more general desire, envie connotes yearning, longing, or craving:
Il peut rester une envie intellectuelle.
There can remain a mental craving.
Cap. 129, Le Figaro - Elle a banni le sucre pendant un an - Part 1
If you think about it, this double meaning of envie makes a lot of sense, since envy is bound up with desire: if you envy (envier) someone, you covet what they have.
J'envie les caresses
I envy the caresses
Cap. 18, Oldelaf - interprète "Bérénice"
But envie isn't always so intense. The extremely common expression avoir envie de doesn't mean "to envy" or "yearn for," but simply "to want," "feel like," or "be in the mood for":
Vous avez pas envie de faire la sieste?
You don't feel like taking a nap?
Cap. 29, Actu Vingtième - Le Repas des anciens
J'ai envie d'une limonade.
I'm in the mood for a lemonade.
There's also the expression donner envie (literally, "to give desire"), which means "to make someone want something":
D'avoir des quantités de choses
Qui donnent envie d'autres choses
To have things in large quantities
That make you want other things
Cap. 4-5, Fréro Delavega - Foule Sentimentale
In English, we have the phrase "green with envy." But in French, one becomes "green with jealousy": vert(e) de jalousie. You can, however, make someone "pale with envy" (faire pâlir d'envie).
Finally, here's a bizarre quirk of the French language: envie is also the word for "birthmark" and "hangnail." What those have to do with envy and desire is an etymological mystery.