Chekhov described a story as "the casual telling of a nuclear experience in an ordinary life, rendered with immediate and telling detail." Telling detail, then, is a fundamental unit of fiction that captures the individuality and uniqueness—the very essence—of what is being described. It doesn't simply inspire an image in the imagination, it also suggest an abstraction, such as meaning or emotion. And it does all of this with brevity.
In The Great Gatsby, F. Scott Fitzgerald describes what it is like for Nick, the narrator, to live next to Gatsby:
There was music from my neighbor's house through the summer nights. In his blue gardens men and girls came and went like moths among the whispering and the champagne and the stars.
The constant music on summer nights implies gaiety. The partygoers are compared to moths, which are light and flitting. There are intimacies at this party, people leaning in close to one another. There is champagne and, no doubt, the loosening that comes with the decadent drink. And there is the very wonder and beauty of the universe. What a staggering environment in which to come and go. Fitzgerald accomplishes this with precisely chosen detail and careful word choice. Telling detail.
In Nelson Algren's The Man With the Golden Arm, Zosh looks out her apartment window at the tracks of Chicago's elevated train:
Some nights she could scarcely breathe for seeing the flat unerring line of cable and crosslight and lever, of signal tower and switch. For the endless humming of telephone wires murmuring insanely from street to street without ever saying a single word above a whisper that a really sensible person might understand.
Zosh is in a wheelchair, but you don't need to know that to feel the confinement in this description, and the limitless stretch of this moment. The telling detail accomplishes that.
Many writers will pile on details in an effort to capture a character, setting, or moment. This is certainly useful in early drafts, as writers don't often hit on the best details right away. Over describing can be a great way to find telling detail. It's important to go back and pare out what isn't necessary so that the good stuff doesn't get lost in the clutter.
The advice to "show" and not "tell" does fit into this conversation, but not in the way you might expect. These two terms—"show, don't tell" and "telling detail"—use two different words. To "tell" is a verb. In fiction, this is one way you can convey information. However, when discussing details, "telling" is an adjective—a kind of detail that is revealing and has great weight. That's a lot of grammar talk, I know, but it's a vital distinction. If you tell, you're going to end up very far from telling detail. Instead, use concrete language that lets the reader experience what is being described. (See Q&A #2 for a more in-depth explanation.)
I'm not quick to modify Chekhov, but I often use other terms for this very reason. I've found that "revealing detail" and "defining detail" substitute nicely. And you might find it helpful to think of telling details in this way until you can easily untangle the two issues.