The soundboard is a slightly crowned section of wood that is designed to reflect all of the sound generated by the strings. It is sometimes metaphorically referred to as "the beating heart" of the piano because of how essential it is for the resonance, volume, and overall sound of the instrument.
In upright pianos, the soundboard covers the back of the instrument while grand piano soundboards are visible underneath the plate and strings:
Soundboards are typically made out of spruce, although manufacturers will often use different species of spruce depending on the region. Spruce is optimal for piano soundboards (and many other instruments) because of how the grain travels in uniformly straight lines, allowing the sound to better travel across the wood's surface.
The crown of the soundboard -- which must be kept under tension in order to properly vibrate and resonate with sound -- can often be indicative of a piano's overall health. Older pianos that have already spent several decades supporting the enormous tension and weight of its parts will often have less resonance, volume, and sustain time than newer pianos. While there are many factors contributing to this degradation of sound, one of the most common issues is the soundboard relaxing over time and losing its crown.