they mean thermal underwear, and the reason they say no cotton is because cotton doesn't insulate worth a damn! if you have to choose between wool and synthetic go for the wool! not only does it keep you warmer but it also has anti fungal properties ( they keep smell away)
as far as the heavy socks is concerned, they are heavier weight than regular socks. again i would say go for the wool, only because the wool socks will wick away the sweat from your feet and prevent you from getting friction blisters. if you must use the bag of synthetic socks that you have be sure to use powder on your feet before you put them on, so you don't have to worry about blisters. and take them off everynight, to let your feet get air
hope this information helps you in some way, and happy hiking!