Historically, a page was a servant employed by an army, lord, or individual knights. In Amtgard, pages are roughly analogous to man-at-arms, in that they are taken by squires. Although this is theoretically a mentor-student type relationship, there are a multitude of reasons that a squire might take a man-at-arms. Everything about pages is custom, so there are no hard and fast rules.
The difference between a page and a man-at-arms: Officially, there is no difference, as neither is officially recognized in the Order of Precedence or most Corporas. Some kingdoms define a man-at-arms as some one who fights, while a page is a non-combatant. Some kingdoms eschew one in favor of the other. Some kingdoms allow squires to take a man-at-arms, and a noble to take a page. In some places, children too young to fight are pages, while everyone else is a man-at-arms.
The term 'Page' also refers to the kind folks that pick up spent equipment for you on the battlefield. Non-fighting pages may retrieve spent equipment, but may not retrieve or carry new or extra equipment.