The fiddle is the father of every American folk instrument(if human voice doesn't count). In country music it has a very symbolical, traditional role, and God knows if country is, among other things, about tradition. That's probably why the fiddle stayed in country music, while practically disappearing from blues and rock (with exceptions like Andrew Bird).
Let's read how Bill C. Malone introduces his discussion about the instrument, in his wonderful book Country Music USA. I'm halfway into the book and I feel I've found a goldmine and the Bible of country scholarship :
The instrument most favored by rural folk, and for a long time virtually the defining instrument of country music, was the fiddle. The fiddle came with the earliest colonists, was soon mastered by nearly every folk group in North America, from the French habitants of Acadia to the blacks of the South, and was then taken to the farthest reaches of the frontier."A great exemple of fiddling in "The farthest reaches of the frontier" is my old post about the Gu-Achi Fiddlers.
The Skillet Lickers - Soldier's Joy (buy) (1929)
The fiddle was a dance instrument, as in this track by the Skillet Lickers (photo above), featuring one of the finest fiddlers of old-time country, Clayton McMichen.
Another great institution is the fiddle contest, that helped a lot of musicians earn fame. A quick Google search will show that fiddle contests are still alive and kicking today.
Bill C. Malone, in his book, makes a short classification of fiddle tunes.
Some of the oldest were of direct Celtic origin, such as "Soldier's Joy", "Old Molly Mare". According to Malone, tunes like "Flop-Eared Mule", the bluegrass classic "Fire On the Mountain" or "Leather Breeches", "although presumed to be American, drew on British airs". Here's a nice version of "Flop-Eared mule" by modern-day Alabama fiddler Jerry Rogers. If you like this, please consider buying the album.
Jerry Rogers - Flop-Eared Mule (buy) (2004)
I'd like to add this John Hartford's version of "Leather Breeches" from his classic progressive bluegrass album Aereo Plain. I took the picture of Hartford below from a post by Nelson at Star Maker Machine.
John Hartford - Leather Britches (buy) (1971)
Other fiddle tunes were American-born, with geographical references in their title, like "Cumberland Gap", "Cripple Creek", "Mississippi Sawyer". Then there were those which commented historical events : "Bonaparte's Retreat" or "8th of July" (...1815, when Andrew Jackson defeated the British at New Orleans, the subject of a famous Johnny Horton song too).
Let's hear of one the most acclaimed fiddle players of old-time country, Arthur "Fiddling" Smith, with the McGee Brothers, palying his lively rendition of "Cumberland Gap". You can find a lot form and about Arthur Smith at Jeremy's Saggy Record Cabinet.
Then, from the Doc Watson Family album, a great "Bonaparte's Retreat" with Gaither Carlton, Doc's father in law, on fiddle (he's the guy on the very first photo above).
Sam & Kirk McGee with Arthur Smith - Cumberland Gap (buy) (1957)
Doc Watson and Gaither Carlton - Bonaparte's Retreat (buy) (1963)