These postures are only suitable for those who are very flexible. I have a friend who had to have the cartilage removed from his knees after years of forcing himself into lotus. If you feel any pain in your knees, or this posture becomes very uncomfortable, then try one of the earlier postures that we looked at.
In the full lotus, the feet rest on the opposite thighs, with the soles pointing upwards (if you have pain in your ankles then stop! and find an easier posture). In the half-lotus, one foot is on the opposite thigh with the sole pointing upwards, while the other rests on the floor, as in the tailor position.
Full lotus is said to be the best position for meditating. The meditator who is able to sit comfortably in full lotus is close to the ground (which, for some reason, seems to be helpful in feeling "grounded"), and is also in a very balanced and symmetrical posture.
Sitting on a chair or kneeling with cushions or on a bench are even more symmetrical postures, but there's less contact with the floor. (If this business of not being on the floor puzzles you, then you need to experience the difference between meditating on a chair and meditating on the floor.)