“Bon courage” is a fairly general well-wishing expression. It can be used in many contexts where the person being spoken to is about to perform a difficult action.

There is no exact English equivalent. Often, but not always, “good luck” can be used in similar situations. The expression “bonne chance” also exists in French, but far more than in English, it carries the connotation that the person will succeed or fail due to purely external factors. In contrast, “bon courage” implies that success will be due to the person’s strength. “Bon courage” also implies some ordeal, some difficulty (though it can be the difficulty of day-to-day life). If there is a genuine ordeal in the person’s path then “bon courage” applies. (Note: I’m speaking as a Frenchman, I believe Canadian French uses “bonne chance” more often.)

A few examples:

— Il faut que j’annonce la mauvaise nouvelle au patron. (I have to tell the bad news to the boss.) — Bon courage !

— Je rentre chez moi, bonne soirée ! (I’m going home, have a nice evening.) — Tu as de la chance, j’en ai encore pour deux heures pour finir ce dossier. (You’re lucky, I have another two hours’s work to finish this task.) — Bon courage !

Je me fais opérer demain. (I’ll be undergoing surgery tomorrow.) — Bon courage ! [“Bonne chance” would carry the implication that there is a chance of failure and is likely to be resented. “Bon courage” implies that this is a difficult moment but that this moment will pass.]

— Le concours commence demain. (My exams start tomorrow) — Bon courage ! OR Bonne chance ! [“Bon courage” denotes the difficulty of the exam; “bonne chance” expresses the hope that the person will be quizzed on topics that they know well.]

— Je déménage demain. (I’m moving tomorrow.) — Bon courage ! [“Bonne chance” would be weird here as there is no luck involved, only a hard task to perform.]

— La première de la pièce dans laquelle je joue est tout à l’heure. (The opening performance of the play I’m performing in is later today.) — Merde ! (Break a leg!) [In French, like in English, explicitly expressing good wishes before a performance is considered unlucky.]

Linguee has more examples of uses of bon courage with official translations.

In persian it means “خدا قوت” (God’s strength). But in English the best word that I found is “Good Luck”.
An example from my French book:

• Hier, on a travaillé les jambes, aujourd’hui on étire le dos.
• C’est bientôt fini.
• Bon courage ! Levez la tête, rentrez le ventre et souriez!

And when a professor use this expression for your works, it means that “Good jobs” and “Keep going”.

That’s all.

My French-speaking friends say «bon courage» when I have some (difficult) work to do and they are leaving me.

I’m a native English speaker fluent in French. I tend to use « bon courage » for more general well-wishes. The etymology of « courage » comes from « cœur », or the heart, and I like to think I’m wishing the person strength. That said, I think this is a pretty particular use – the other answers have covered the general stuff wonderfully !

“Be of good heart” is correct, remembering that both phrases (French and English) comes from a time when the heart was considered the seat of resolve and impetus, not just romantic passion. Consider the adjective “hearty”. Or the song, “you gotta have heart (all you really need is heart)” from Damn Yankees. In English, we now use it mostly for race horses and sports. The French still use it more generally.

Example: last week, as I (middle aged, rotund and female) was chugging slowly up a longish hill on a bike, I answered the “ça va?” of an older gentleman with a laugh and a “so-so” hand signal. He replied, “mais vous êtes courageuse! Bon courage“. This wasn’t about my “courage” in the sense of fear or doing anything death defying or extreme. It was a “you go, girl,” a “thumbs up”, a “good for you, you can do it”.

A litteral equivalent is probably "may the force be with you" "use the force"… Jamaicans say "Strength and Unity" which is fairly equivalent too, because it implies mental strength and unity.