Contrary to what Comair has said, they are NOT the same.
Regional airlines are independent companies that operate as free enterprises. They compete against each other for contract flying offered by the major airlines.
Skywest is owned and operated by SkyWest Inc, which also provides contract flying service to United, Delta, and Midwest Airlines. They bid for flying that is available and are awarded contracts based on price, service, and reliability.
Chautuaqua airlines is simply one of Skywest's competitors, along with ExpressJet, Mesa, Comair, and many many more. The reason you generally see "United Express" or Delta Connection" painted on the planes is more for customer assurance than anything. If someone buys a ticket on Delta, they prefer the airplane to say Delta on it. It allows a bit of transparency between the operations of the companies. If you try to fly from HoDunk, Alabama to Middle-of-Nowhere Idaho, odds are there will not be a direct flight. Odds are also that there will not be a mainline flight that goes to either of those two places. Delta cannot sell enough seats to justify putting a 737 into HoDunk, so they contract the route to a company that operates smaller aircraft with less capacity. That way they only need to sell 40-45 seats to be cost-effective.
That flight will usually leave Ho-Dunk, fly to a Delta hub (Like Atlanta, then a mainline flight will take the passengers from Delta hub to Delta hub (SLC in this example) where they will connect onto another contracted regional company to fly into Middle-of-Nowhere. Delta cannot afford to fly its airplanes to all these places, so smaller planes are hired on a longer-term contract basis to provide it for them. Delta gets the revenue from the ticket sales and the regional airline is paid on a fee-per-departure basis.
I fly for SkyWest.