To experience a bit of live, impromptu Spanish guitar, namely Flamenco, head to this cuevas (caves) bar on the outer wall of the Plaza Mayor. It is relatively easy to find, but a gem that doesn't get frequented by many tourists. The atmosphere is dusty and romantic, and perfect for traditional tapas with a jug of sangria. Don't come too early or you'll miss the older men who play there on a rather irregular schedule, but usually arrive sometime after 10pm.
15, Cava de San Miguel.