What is a rhumb line?
a rhumb line, rhumb, (/rʌm/) or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north


From: Wikipedia

If continued, all rhumb lines except the ones following a  track of 90 or 270 degrees end up in either the North or South Pole.

If continued, all great circle lines end up back where they started from.

As for the part of the question about constant track direction: track is defined by ICAO as


  The projection on the earth’s surface of the path of an aircraft, the direction of which path at any point is usually expressed in degrees from North (true, magnetic or grid).


So, to maintain a constant track would mean that North would always be in the exact same direction from you as you move on. This would eventually take you to one of the poles, unless of course, you were maintaining a track of 90 or 270 degrees.

The picture above shows a constant track of about 75 degrees.