| There are many variations, subtleties, and nuances. And if you hang around here long enough, you will learn that anything stated as a "fact" will be disputed by someone!
But from a very basic point of view (using a right handed approach):
The fingers of the right hand are used to pluck the strings. Usually the index and middle fingers are used, although some players use a pick (or plectrum.)
Typically the fingers pluck only one string at a time. Although chords can and are played on the bass, typically only one note sounds at a time.
These same fingers, or others, can also be used to stop the strings from vibrating after they have been plucked. This shortens the duration of the note.
The fingers of the left hand are used to depress the strings. The string is stopped at the fret where it is pressed, so essentially the string is made shorter when a note is fretted. This controls the pitch of the note that is heard.
Typically one finger is used to depress the string - which one depends primarily on what notes preceed and follow the one being played. The other fingers of the left hand may contribute by muting the strings which are not being played at the moment.
Of course there are exceptions to all this - these are not "rules" but just a description of what happens "most of the time." | 
