What are the 2 most common methods of "hands-off" coffee brewing? 

Regular coffee makers, like your Mr. Coffee, are “drip” brewers. They work by heating water to near boiling and routing the steam to a drip area, which disperses water over coffee grounds. The finished brew filters into a pitcher over a warming plate. Depending on the quality of the machine, your coffee may taste either delicious or a bit scorched, but the method is hands-off and convenient.

By contrast, French press coffee is usually strong, with a rich, earthy aroma. A French press doesn’t require electricity or disposable filters to brew, just hot water and coffee grounds. The grounds steep directly in hot water, and are then filtered out when the user presses down the plunger. French press is probably the second most hands-off brewing method, after drip.
Both "drip" and "French press" are the most common hands-off coffee brewing methods.