Like most western pop and traditional music, country uses close harmonies, meaning notes that are in within the same octave. What makes for great country harmonies, however, other than the fact that many country legends grew up singing together, is their ingenuity within traditional song structures. Other than that it shares the fundamentals of western harmony, such as how the first, third and fifth scale degrees function in a chord, there are no set rules for harmony in country music. It does have some harmonic tendencies, however, which you can learn to give your singing more of a country sound.
When you are harmonizing with a lead vocalist, generally you want to be singing one of the other notes in the chord. Thirds are the most agreeable sounding harmonies, so if the lead singer sings the root, sing the third of the chord. If he or she sings the third, sing the fifth or the root.
One tendency of country harmonization is the ringing out of the root note of the key in which you are singing, sung one octave higher. Most commonly this will occur when you are playing the IV (i.e. a C when you are playing in G): One singer sings an E, the third in a C chord, while the other sings a G which is a third higher, the five in a C chord which is also the tonic.
You can add to the country feel of these kinds of harmonies by throwing in suspensions and anticipation notes. For example, if you're still playing a song in G, then when you move to the V chord, which is D, have the high note slide to the G while the lower voice sings D. The interval will be a fourth, and the high note anticipates a resolution back to G.