What is this pattern at the bottom of the default cube? Start Blender, hit Tab to enter edit mode, orbit to see the underside of the starting cube. The bottom face is marked with a grey pattern, unlike all the other faces of the cube. What does this mean? I haven't seen this pattern in any images or videos in tutorials.
It is the 'active face', which is the face that was selected last. 

This can be useful for actions like transforming, where if you change the 'Pivot Point' to 'Active Element' you can scale towards or away from the 'active face'.

'Vertex' and 'Edge' selection modes also have 'active' elements and will display the last selected element lighter than the others. Object mode also does this when you have multiple objects selected, which can be used to (among other things) copy settings from the active object to the other selected objects.