Tempo: music should be played calmly and slowly, but slightly faster than lento. It could also be described as leisurely, without rushing, at ease. The music is played slowly, but it is less deliberate than lento.
The best example to illustrate adagio is Adagio in G Minor by Albinoni, as shown in the video below. While the piece is slow, it drags less than lento. We can feel more of a movement in the music than in the lento pieces.
The second movement of Beethoven's Pathétique Sonata is another example of adagio. The music is played at a leisurely pace but it doesn't feel like it drags.