Introduction: Hot Grog! Avast Ye!
History (read: Wikipedia) tells us that Grog was first introduced in 1740, when Vice Admiral Edward Vernon of the British Royal Navy ordered that his sailors' rations of rum be diluted with water or "small beer" (a weak beer). To cut down on the foul taste of the (typically filthy) water, a bit of lemon juice was added. Et voila, Grog.
Alternatively, rum with water, sugar and nutmeg was known as "Bumbo" and was more popular with pirates and merchantmen, because it tasted better than the Royal Navy Grog. For more on grog history the Wiki article is a good place to start.
Modern Grog is a bit of mixture between ye olde Grog, Bumbo & a Hot Toddy and, served warm, is an awesome cold-weather drink which I actually prefer to a toddy. It also gives you a perfectly legitimate excuse to talk like a pirate for a bit. BONUS.
This is my preferred version.
1 shot rum of your choice, dark or white (I prefer dark)
1 tsp sugar
Squeeze of lemon or lime
Boiling water to fill a mug
In a mug, mix together the rum, sugar and lemon until the sugar has dissolved, and then add the cinnamon stick. Add enough boiling water to fill the mug, or to taste. Since the cinnamon stick has done enough to diminish Grog's manly historical image, you might as well add a slice of lemon or orange and drape it langorously over the rim of the mug to decorate.
Historically accurate Grog: In a tankard, mix cheap rum with fetid water & a squeeze of lime and feel grateful that you haven't got scurvy.