When I play one note in piano, and play another after that, the latter makes the first silent. Does a particular note stops when another note is played , especially in piano?
In general, you're theoretically supposed to stop playing a note at the end of its nominal duration. In most music another note is written at exactly that point, but there's no rule that requires this. So if there's a piece with a metronome mark setting the quarter note at sixty beats per minute, you play the quarter note for one second, regardless of what comes after it. Theoretically.
I say "theoretically" because musical interpretation and expression may lead a performer to shorten or lengthen a note slightly, or more than slightly. Sometimes this is in response to an articulatory or expressive mark made by the composer, and sometimes it comes from the performer's own sensibilities.
When you're playing the piano, each note you play stops sounding when you lift the key, unless you're using a pedal (other than the soft pedal).