Playing country music on guitar involves a lot of open chords, meaning we're using chord shapes that have a lot of open, unfretted notes. Some people might call those "cowboy chords".
They are some of the most basic chords, but they sound really good on acoustic guitar because they have a lot of sustain and a lot of volume. The more sustain and volume you have, the more control you have over the instrument. Here's a little string of chords that would demonstrate these fingerings.
That gives us a chance to pick out some melody while still keeping a rhythm. I guess the term "cowboy chords" comes from a bunch of cowboys sitting around a campfire late at night on the wagon train and obviously, they can't be dragging around too much more than a guitar, maybe a fiddle if they're lucky, a harmonica. Since they're so busy during the day, at night they are really only going to know some of the chords; hence the name "cowboy chords".