What is the meaning of the alternative rock band's name Depeche Mode?

It is a common mistake to assume that the name Depeche Mode is purely a translation from the French for 'fast fashion'. Where this error started is hard to establish and when music sites such as the All Music Guide still get it wrong (also used for Biographies on the now defunct CDNOW), it is easy to see why this error still perpetuates across hundreds of sites.
depeche means dispatch (or update, message or news), mode means fashion.

So Depeche Mode means Fashion News or Fashion Update. The confusion arises from someone remembering their high school French: the verb 'se depecher' means to hurry, which is probably where the 'fast' originates (thanks to Lawrence Koch for explaining the French translation and the source of the error). So, Depeche Mode is named after the magazine and that name roughly translates as Fashion News.

Another source:

When explaining the choice for the new name taken from a French fashion magazine, Dépêche mode (from French dépêche that means here "dispatch" (from Old French despesche/despeche) or "news report", and mode that means "fashion"), Gore said, "It means hurried fashion or fashion dispatch. I like the sound of that." But, in French, the real and only meaning of the magazine's name (and hence the band's) is "Fashion News" or "Fashion Update"

https://en.m.wikipedia.org/wiki/Depeche_Mode