A compass needle in the northern hemisphere points north. Is the end of the needle pointing to the north a “north pole of the magnet” or a “south pole of the magnet?” How do you know?

It depends on how you define "north." If you define the Earth's north pole as "north" then the "south" side of a magnet will face it. If you define the magnet's north pole as "north" then the Earth's magnetic north pole is going to be defined as "south." 
The best way to visualize this is to think of magnetism as a consequence of flow. With only a recently discovered exception involving spin ices, there is no such thing as a "magnetic monopole." Each magnet has a north (outflow) and a south (inflow). Think of magnetism as a fluid that needs to flow in and out of a magnet, and it can't accumulate in a magnet. All outflows (Norths) go to an inflow (South).
In this way, it might be easier to visualize how the magnetic "north" of Earth would oppose the magnetic "north" of another magnet.