A progression is the chords of a tune in order. For example CM7 F7 G7
An arpeggio is when you play the notes of a chord one at a time i.e., C, E, G, B is a CM7 arpeggio in the root postition. Walking is when you play notes against the progression, typically quarter notes, such that they relate to the progression and hopefully sound good.
Like Art says, a decent grasp of theory is very helpful when learning to walk.