Felt, felt, and more felt!

Just the same way the strings vibrate the soundboard, they have a tendency to vibrate the whole piano and all its parts. That wouldn't sound too pretty, so there is just as much done in the construction of a piano to PREVENT noise as there is to cause it.
Felt in a grand piano
Grand and upright pianos are full of pretty
felt that comes in a variety of colors.
The best way is to insulate all the parts that might rub together or rattle against one another, especially where there are moving parts that have to sit right up against each other. Felt or leather is used on all the little bits of action that have to hit each other or pivot to move the hammers and dampers, so that the parts don't click together or make scraping noises as they move. Felt is also used in major joints where large parts of the piano fit together and might vibrate against one another. If you look closely, you can see brightly colored felt in lots of places, especially in a grand piano with the lid opened up. It is all to prevent unwanted rattles and vibrations.

Felt dampers sit on the strings
A felt damper sits on each set of strings
until the note is played
Another VERY important place where felt is used is in the dampers. These are mutes that sit on all but the very highest treble strings. They keep the sound board from vibrating the rest of the strings when one string is sounded. If it weren't for dampers, playing one note on a piano would cause the rest of the notes to ring and make a strange string echo noise. When a key is played, the action lifts the damper off that one string and holds it up for as long as the key is depressed. If a damper is malfunctioning and doesn't lift, the string is unable to vibrate and just makes a weird dull "thunk" sound when the key is played.