A bass run is a sequence of bass notes which occur between two chords. Many bass runs begin on the root note of the first chord and lead to the root note of the second chord. In the following Country rhythm part, bass runs are used to connect all the chords.
Pay careful attention to the fingering in this example. The D chord in bar 5 is simply the C chord shape moved up two frets, so that the first finger is at the 3rd fret on the second string (a D note).
The 4th finger remains on the first string throughout the example and is only lifted for the final D chord (bar 8) and the bass run leading up to it.