Anytime in written music that there are multiple staffs, they are all played simultaneously. Exactly who plays/sings what part will depend on the context of the music. Typically, in vocal music with two staffs, the upper staff (with the treble clef) will contain the female parts (Soprano and Alto), while the lower staff (with the bass clef) will contain the male parts (Tenor and Bass).
You can check that this is the case by noticing whether there are two different notes on each staff (though sometimes the two parts will coincide on the same note). This type of 4-part vocal arrangement is very common, and is often called an S.A.T.B. arrangement (for the 4 voice types). Typically, the Soprano (the upper female part) has what you would think of as the normal melody (which is also the part that non-musicians would sing), while the other three parts are singing various harmonies (if you aren't used to this idea, this means that, in addition to the melody, you have three other independent musical lines being sung simultaneously).
In addition, as Rockin Cowboy says, keyboard instruments can play both staffs together. The right hand usually plays the top notes, and the left hand plays the bottom notes. Alternatively, they can improvise an accompaniment that matches the same chords, but doesn't necessarily have to stick to the exact vocal lines. Chordal instruments like guitar have to deduce the chords from the music, unless they are written out above the staffs. Other instruments are free to play whichever part they want (as long as it fits their instrument's range) although it's usually a safe bet to double the melody (the soprano line) unless you're playing a bass (in which case you would double the bass line -- the lowest male part on the bass staff).
In the more general case of ensemble or orchestral music (where you could easily have over a dozen staffs at once) the staffs will usually be labeled as to which instrument is intended for each staff.