With alternate picking you pick up-down-up-down.... the whole time, even when going from one string from another. Now, with economy picking, you're trying to minimize your movements(hence the term economy). I'll explain with a 3 note per string ascending scale. It's would be something like this: down-up-down-down-up-down-down-up-down-down... As you notice, between every 3rd and 4th note, you go from one string to another, but you downpick both those notes.
So basically a economy picking is picking notes from adjecent strings in one continous motion.
arpegios: chords with each notes picked seperately in a descending or ascending order
Economy picking is very often used to get those lightning speed arpegios.
Legato's: I think this means "smooth"
Basically it means you play a bunch of notes without picking them. Thus, using hammer-ons and pull-offs(and maybe even slides).