Exactly What It Says on the Tin, a cousin who hails from the rural parts of the nation. This trope usually comes into play on shows set in the suburbs or the city, and it usually takes the form of the cousin (or cousins) coming to visit or one of the show's main characters going to the country to visit the cousin. Either way Hilarity Ensues.
Doesn't necessarily have to be a cousin; it could be any relative. Cousin is just being used as a general term here, since it is somewhat of a common phrase.
Compare Country Mouse; while that trope usually applies to characters who are naive and central to the cast, this more applies to a character with more or less any personality (though stereotypes applying to the country will usually be apparent) and is at most a Recurring Character.
See also The City vs. the Country.
The story of the city mouse and the country mouse (a theme/idea of Aesop, written into a poem in 1948 by Lauri Pohjanpää). The story: the city mouse visits the country mouse. Country cousin offers the best it has, but City cousin sniffs its nose at everything and goes on about how everything's better in the city. So Country cousin later visits City cousin. City cousin lives in a castle, where the food is marvellous, but the mice have to be ever alert and run and hide, and getting a treat is a tough task. Country mouse thinks it's better to eat humble food and live at peace, than to sneak around and eat handsomely, and returns home.
In I Love Lucy, Lucy's Cousin Ernie comes from rural Tennessee and quickly overstays his welcome. Getting him to leave takes 2 episodes and, later in the series, Ricky and Lucy see him in Tennessee. Cousin Ernie was played by Composer and Actor Tennessee Ernie Ford.
In Married... with Children, Peggy's family consists of country hillbillies who live in the middle of nowhere. They appeared in a few episodes where the Bundys visited them, and when Katey Sagal took a break from the show while pregnant, the character went to Wisconsin to stay with her family.
In Justified, Raylan's late mother Francis descended from Kentucky hill-folk, much to Arlo's embarassment. His blood ties to Mary, the matriarch of Cope's clan, end up saving his life in "Kin".
The Flintstones had an episode in which Fred's country cousins (who just so happen to be over-the-top Stone Age hilbillies) come to visit for a weekend...and then continue to stay for months afterwards, thinking that Fred really means it when he says that he wishes they could stay longer.
On SpongeBob SquarePants, Plankton call his relatives to help him get the Krabby Patty formula. He expects them to be Evil Geniuses like him, so he's shocked to find that they're all hillbillies.
Spoofed in The Simpsons, where Homer's country "cousin" (actually only their dogs are related) is way more sophisticated than him, whereas Homer claims it's the other way round, adhering to the stereotypes concerning country people.
Inverted in South Park with Kyle's cousin Kyle, from New York. He's very much the stereotypical Jew/nerd (nasal voice, good with math and money, and heavily allergic to EVERYTHING). In the end, he calls Kyle and his friends hick jocks.
Another take on the same premise was two versions of Little Red Riding Hood (or rather, Red Hot Riding Hood) where the country cousin visits his city-cousin wolf to get away from his countrified Abhorrent Admirer, Red, only to fall for the city version of Red. Once the City Wolf takes him back, he then has the same reaction to Country Red.
In My Little Pony: Friendship Is Magic, Applejack was a Country Niece to her Aunt and Uncle Orange in Manehatten, whom she lived with for a brief period as a filly, but quickly found that she missed her more immediate family, and the farm.
Inverted with Babs Seed, who visits her cousins Applejack and Apple Bloom in the rural Ponyville from Manehatten.