Allegro means "fast". Allegro pieces are typically played in a lively and upbeat manner.
Adagio means "slow". Adagio pieces are typically played in a relaxed and stately manner.
Here is a table summarizing the differences between allegro and adagio:
| Characteristic | Allegro | Adagio |
|---|---|---|
| Tempo | Fast | Slow |
| Mood | Lively and upbeat | Relaxed and stately |