Plain English Meaning
MOAs are chunks of sky reserved for military training. When they are switched on, fast military jets may be doing aggressive maneuvers inside. You can legally fly through one as a VFR pilot, but it is much safer to call ahead and check what is going on first.
Full Definition
Military Operations Areas (MOAs) are blocks of airspace with defined vertical and lateral limits established to separate certain military training activities from civilian IFR traffic. When a MOA is active, military aircraft may be conducting maneuvers such as acrobatics, formation flying, or high-speed intercepts inside it. VFR pilots are not prohibited from flying through an active MOA, but they are strongly advised to contact the controlling agency before entering. IFR traffic is routed around an active MOA by ATC unless can be assured.