Let's consider the piano version of Moldau/Vltava from Smetana, the first two measures for example:
What should be the duration of the blue circled notes, as expressed in units of an eighth note:
Should it be less than an eighth note, or equal to an eighth note?
Is it staccato or something else?
If this is less than an eighth note, then would that mean the right-hand circled notes do not "fill" the rest of the measure? (and if so, why isn't there some sort of silence added to the score?)
The circled notes are an accented (loud) staccato, so the sound of the note will not fill the entire time allotted to those eighth notes. However, the silent part of the sound is built into the meaning of staccato, which is why it's not notated as, say, an accented sixteenth note and a sixteenth rest. It's understood that a staccato note will include some portion of silence.
This still begs the question of why an eighth note is used, and that has more to do with the interpretation — the musical feel — of the passage. There is (rather, can be) a felt difference, for example, between a staccato eighth note and a sixteenth-note / sixteenth-rest pair. However, that difference is beyond the realm of description and rather one of experience and feeling.
In addition to Aaron's great answer, it's worth mentioning that staccato literally means "separated." (The common meaning of "short" really only tells half the story at best.) As we're interpreting note lengths of these markings, I think it's important to understand what the marking actually means; I've seen whole notes marked staccato, and they definitely aren't played short!
Some musicians, especially in competitive settings like large marching bands, quantify specific durations for these markings; I often see staccato listed as "50% sound, 50% silence." I guess this works in some sort of mechanical way where a judge's score is your primary concern, but I invite you to be far more fluid with your own interpretations.
The difference between staccato notation (which indicates the total length of both note and following separation) and an explicit sequence of note and rest is not just that the amount of separation is open to interpretation with staccato. It is also that a staccato note does not indicate the end of a musical phrase: without a following rest, the next note will typically be considered part of one phrase with the staccato note, implying a continuation of the musical tension and cohesion across the note separation.
I agree with all the previous answers, but would like to add that this piece is a piano transcription of an orchestral work, and that the chords in question are played on pizzicato (plucked) violins. That should give you an idea of how they should sound: probably shorter than a semiquaver (16th note).
