i-adjective
The main everyday sense: a person, schedule, workplace, season, etc. has many things to do and little free time.
今日は仕事が忙しい。
I'm busy with work today.
年末はどの店も忙しくなる
Every shop gets busy at the end of the year.
i-adjective
Describes movement, attention, or manner that is constantly shifting or hurried. Less central than the 'busy with work' sense, and often used adverbially as 忙しく.
小鳥が枝から枝へ忙しく飛び回っている。
The little birds are flitting restlessly from branch to branch.
彼は話しながら視線が忙しかった。
As he talked, his eyes kept darting around.