Allegro means "brisk or lively (generally fast)". It is a musical term and obviously has other meanings and contexts. Marcato is to be "accented" or in a "marked style" - usually a driven rhythmic thing with notes or beats 'stressed'. Allergo Marcato is the "tempo and style" marking for this movement from a Prokofiev symphony (#5/2nd movement). Not real clear but that's it....that is what it means.
Playing a difficult piece like this and making it "sound" like the composer intended and in his style (think of an impressionist doing a voice - and think about how it is as much the nuance and stresses that a performer puts into the voice that makes it come alive - well it is here) is something musicians learn over time. Same instrument. Same notes on the page. But a different voice for nearly every composer.
This is hard work and very interesting visually. You don't have to hear it to understand it is "marcato". The performance, by high school players, is amazing....and I stress that.