This tip is probably most useful when you're working in a table, although you can use it to reorder paragraphs outside a table, too. Let's say you decide you want the third row of a table to be the top row. Just click within the third row, hold down [Alt][Shift] and press the up arrow key twice. Each time you press the arrow key, Word will move the row up one. You can select multiple contiguous rows to move them as a block, and you can use the down arrow key if you want to move text down instead of up.
Using this shortcut gets a little tricky if you're moving big pieces of text outside a table. It's easy to lose track of what's being relocated where, and you might find it easier to take a standard cut-and-paste approach in those situations. But when the text is small and manageable, the shortcut is great. For example, if you need to move an item up or down within a bulleted or numbered list, you can just click in the item's paragraph and use the [Alt][Shift] and arrow key combo to move the item to the desired spot.