"Nautical mile (n mi) – one minute of arc of a great circle"

Is a what of a what?

One minute of arc is 1/60 of a degree. One degree is one "hour" in terms of measurement. (You also can have one second of arc, which is equal to 1/60 of a minute of arc.)

A great circle is any circle that passes through the center of a sphere (think of it like cutting an orange into two equal halves - once you cut it in half, open the orange and look at the two halves; they are each circles). Nautical miles are measured using the same thing, but making imaginary cuts through the center of the earth. The reason the cuts have to go through the center is otherwise you could make really large or really small circles which would mess up your ability to use that as a measurement. Compare cutting the earth horizontally at the equator - two approximately equal pieces - to cutting it horizontally at the arctic circle (you'd end up with a tiny little piece at the top and then the entire rest of the earth, with a much smaller circle being the "cut"). Making the two pieces be the same size (by passing through the exact center) guarantees that all of your circles are the same size.

http://i.imgur.com/cn74Tnf.png

And this? That's where I got really confused.

Where did the +30 come from? What is this?!

Can't see the whole page but looks like it is because one of the points you are measuring is on the far side of the north pole, so you have to add a full 90 degrees to get from equator to north pole, then add an additional 30 degrees past that to get to point "B". (Since the North Pole is at 90 degrees N, 90-60 makes 30 more to get to 60 degrees N on the far side.)