The fundamental way a piano produces sound is by vibrating strings. As with stringed instruments like guitars, violins and harps, the pitch of a string changes with the length and tension of the string. Unlike a guitar however, a piano contains individual strings (in practice, groups of two or three) for each pitch you might play, so there is no need to change the length of a string as you play it. To play any note, you simply press the key that causes hammer to hit the strings that are tuned to that note.
How does pressing a key cause the hammer to strike a group of strings? This task is performed by the action, which is a complicated connection of wood or plastic parts that tranfers the downward motion of pressing the key to the striking motion of the hammer. (see an animation here: http://www.musicplay.com/action/action.html). This striking motion is the defining feature of the piano and separates it from its predecessors like the harpsichord and clavichord: No matter if you tap the key or hold it, the string is struck briefly by the hammer, which then immediately pulls away from the string to let it vibrate (though it will be damped once the key is no longer depressed). Futhermore, the volume depends upon how hard you pressed the key. Before this invention strings were either plucked, or struck and held in specific places to both sound the string and create the correct pitch. Neither way offered very much dynamic control.
A few strings vibrating in air does not create a very loud or robust sound, which is why pianos and their precursors all have a soundboard. This is a large piece of slightly concave wood that “collects” the vibration of the strings and retransmits them with more volume (like the body of a violin or acoustic guitar).
Those are the basics of how a piano creates and transmits its sound. It is also important to stop strings from vibrating, which is why all strings are damped except when the key associated with them is depressed. Much of the innovation in piano-making has involved more sophiticated ways to control the dampers, mainly with the use of pedals. Some prevent any dampers from being applied, or prevent only the bass dampers from being applied, or leave strings to vibrate if their keys were depressed when the pedal was depressed.
There are many more supporting elements of the piano, and the instrument has undergone significant change since it's original invention in Italy over 400 years ago. To learn more about how a piano works, check out:
http://www.frederickcollection.org/works.html
http://en.wikipedia.org/wiki/Piano
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