What does "keep the black dog at bay" mean?

Black dog is an oft-used phrase to mean depression.

Here's a link that attributes the phrase to Winston Churchill, but I suspect he didn't invent it. I like the way they described it though:

"Black Dog" was Churchill's name for his depression, and as is true with all metaphors, it speaks volumes. The nickname implies both familiarity and an attempt at mastery, because while that dog may sink his fangs into one's person every now and then, he's still, after all, only a dog, and he can be cajoled sometimes and locked up other times.

This paper (PDF, sorry), claims several references in the 1800's. Interestingly, the way they describe it (being ghostly and following people around) makes one wonder if it isn't ultimately related to the English folklore character Black Shuck.

T.E.D's answer is great. Just some minor things that might complement his:

None of Cambridge, Oxford, Merriam-Webster, and even Urban dictionary registers this phrase.

Actually, I've found an entry for black dog on Urban Dictionary.

What does "keep the black dog at bay" mean?

"Keep something or somebody at bay" means to prevent something or someone unpleasant from coming too near you or harming you. "Black dog" means a bad mood, characterized by anger, depression, or a mixture of the two. So the whole sentence means to prevent the bad mood from harming you.

I wonder whether “black dog” is used as a pun with “winter blue” in the headline.

I think "winter blue" just means the same thing as "black dog". "To beat the winter blues" and "to keep the black dog at bay" beautifully echo with each other in the text.

Is “keep the black dog at bay” a well-established idiom?

I don't think the expression as a whole is a well-established idiom. For example, we can use "keep the winter blues at bay" and "beat the black dog" instead.

"Black Dog" as a metaphor for depression is attributed to Samuel Johnson, and popularized by Winston Churchill.

From Wikipedia (Samuel Johnson entry): "On 17 June 1783, Johnson's poor circulation resulted in a stroke[162] and he wrote to his neighbour, Edmund Allen, that he had lost the ability to speak.[163] Two doctors were brought in to aid Johnson; he regained his ability to speak two days later.[164] Johnson feared that he was dying, and wrote:

The black dog I hope always to resist, and in time to drive, though I am deprived of almost all those that used to help me. The neighbourhood is impoverished. I had once Richardson and Lawrence in my reach. Mrs. Allen is dead. My house has lost Levet, a man who took interest in everything, and therefore ready at conversation. Mrs. Williams is so weak that she can be a companion no longer. When I rise my breakfast is solitary, the black dog waits to share it, from breakfast to dinner he continues barking, except that Dr. Brocklesby for a little keeps him at a distance. Dinner with a sick woman you may venture to suppose not much better than solitary. After dinner, what remains but to count the clock, and hope for that sleep which I can scarce expect. Night comes at last, and some hours of restlessness and confusion bring me again to a day of solitude. What shall exclude the black dog from an habitation like this?[165]"

The key part of the expression is "black dog" and not the "keeping black dog at bay". "Black dog" means depression or sadness. Thus, "keeping the black dog at bay" means one is keeping away the depression that attempts to consume him or her.

Cambridge dictionary defines it here
Collins dictionary defines it here