What does the power rating of 10W mean?
Neither. You're confusing energy and power. Power is energy per unit of time, and one watt is one joule per second. So if you have a 10 W device, that means, that it's consuming 10 J each second. 

Now, the issue with electric energy is, that joule, as a unit, is quite small. Therefore, we use larger units, such as wat·hour or kilowatt·hour. Note the multiplication there! Hour is 3600 seconds. So one watt·hour is 3600 J, and one kilowatt·hour is 3600000 J. 

So your 10 W device will spend 10 watt·hours, if it works for one hour, assuming that it always uses same amount of power.

Now, the confusion comes, when people start dropping the time, and start referring to kilowatt-hour as kilowatt. Don't do that, and be careful when others do, since they might now understand what they're talking about.