This. We bought a small cabin in Virginia on X Hollow Road (where x is a family name). This Hollow, or holler, is exactly what P-man describes.
Holler is a term that arose in Southern Appalachian where there are many small valleys that cut into and/or are tucked between the generally SW to NE run of the mountains. Many of the mountains of SW Virginia and NE Tennessee are long, relatively narrow, ridgelines. Zoom on in to that linked topographic map and you see hundreds of little hollows.
What’s the difference between a holler/hollow and a canyon, gulch, gully, or ravine?
Where it’s located, obviously.
I lived in one in extreme eastern Kentucky some years ago, not far from Dog Fork Road and about ten miles from the Ashland power plant which was eerily lit up at night, and which we used to refer to as “Christmas In Hell”.
Correction: “Christmas In Hell” was the big refinery complex which greeted nighttime travelers crossing over into Kentucky on I-64 from West Virginia.
Them’s my old stomping grounds. If Washington Irving had lived there, he would have written “The Legend of Sleepy Holler”.
A holler is much smaller than a canyon. A canyon would have a river running through it, a holler would have a small stream that feeds in to the river that forms the valley. Using Virginia as an example, the Shenandoah valley is a wide valley with the Shenandoah river running through it. The mountains that form the valley are dotted with hollers, each of which has a small stream feeding into the Shenandoah. Many of these hollers have a few houses along them.
A gully is the result of water from a heavy rain rushing down a hill. We would get them on the dirt road going downhill from our house. Hence a heavy rain was sometimes called a “gully washer”.
To the east side? To that DE-luxe apartment in the Sky-y-y? 
Valley is to holler as tree trunk is to tree branch (or maybe twig).
In the big flat flood plain of a wide valley, where two major rivers join, that’s where you generally expect to find a city. But follow either of those rivers back upstream and you’ll likely find that they branch, then branch again, multiple times. Follow one of those little branches back into the woods, far off the beaten path, and you’re in a holler.
Obviously, we’re talking about a rural area. But it’s more than that; it’s a poor area too. The land in the holler is likely to be cheap because #1 there’s less demand being so far away from the main roads or the city, and #2 there’s not so much daylight, so it’s hard to grow crops back there, and #3 when the river floods it tends to wash the topsoil down into the valley. So the valley generally ends up with rich soil and the hollers end up with rocky soil. I’ve also heard such places called a “mailbox community” because so many of the residents just scrape by on a disability check from the government in their mailbox once a month and very few people there have jobs.
In a nutshell: rural, poor, a long way from the city, low property values, very few jobs.
We weren’t exactly poor, but small farms don’t bring in a lot of money. My parent’s farm, which is about 50 acres, lies between I-81 and US 11. Development is creeping closer, and it wouldn’t surprise me if speculators have their eyes on it. It would be kind of sad to see the farm be built up when my parents are gone (and they’re in their eighties now), but realistically that’s probably what’s going to happen to the holler I once called home.
Canyon is generally found in the American SW, from the Mexican-Spanish cañon, which described a narrow, deep gorge with a river flowing through it. Derivation is from Spanish cañon, reed or pipe, from Latin canna; cf. canna (lily).
Ravine in America is more generally found in places that were settled by English, as the word has been in the English language in its current meaning since the mid-1700s. It comes from the French ravin meaning gully, from Old Fr. raviner, meaning to pillage, sweep down, cascade. The original derivation is from the Latin rapina, meaning forcible sudden plundering. A ravine is generally smaller than a canyon, has steep “V” sides, and may not always have a flowing watercourse.
A gully is an English term meaning a channel in earth made by (fast) running water. Origin is not certain; possibly from Middle English golet (water channel). A gully is a small feature, no where near as big as a ravine. A heavy rainstorm can be called a “gully-washer” for the fact that it fills and washes out the debris accumulated in a gully. Growing up as I did at the base of the Sierra Nevada (east side), that term was common for the sudden thunderstorms that produced flash-flooding in the gullies, washes, and small canyons of the area.
Gulch is a nasty woman who takes people’s small dogs from them. 
Then you creek/crick.
Cx: Then you have creek/crick. Ie, one more example.
As a boy from suburban Southern California, I’ve been conditioned to think of clod-hoppers as a kind of work shoe, a thick-soled, ankle-length lace-up leather boot, suitable for walking among broken up terrain (usually with a bunch of clods of dirt).
As a description for a person, it would seem that a clod-hopper is a farm worker, accustomed to “hopping” among the clods of earth found in a freshly plowed or planted field.
That’s DIY etymology, btw. Don’t take it for authoritative philology.
I’ll get you for that joke, my pretty, and your little dog, too!
Ain’t no Cowboys in the holler.
Appalachia does have canyons, but no gullies or gulches. Canyons are steep sided narrow river valleys where the mountain goes straight into the river with no flood plain. They differ from hollows in that they are larger and have essentially zero settlement due to the harsh topography. Cheat Canyon and Blackwater Canyon are examples. We also have gorges which are similar to canyons and narrows which refer to the river in a canyon that moves faster due to a narrowing of the Canyon. You can google map wv-72 north of Rowlesburg to see Cheat Narrows and Cheat Canyon has no road, but extends between Albright and Cheat Lake.
“Holler” is featured in this wonderful David Bromberg piece which I just love: Moonshiner - YouTube