早く：can mean either (1) early; or (2) fast/quickly (mostly used to describe the speed with which one performs an action). When used to mean fast, it can also be written as 速く.
- 早く来てください。(Please come early)
- 疲れていたので早く寝た。(I was so tired that I went to bed early.)
- 彼女はなんて早く泳ぐんでしょう。 (How fast she swims!)
- 彼は速く走れる。(He can run fast.)
急いで：means "in a hurry". It comes from the verb 急ぐ, which means "to rush, be in a haste." Unlike はやく, it has an emotional nuance (you do something fast/in a hurry because of some pressing (emotional) matter).
- 彼は急いで空港にいった。("He went to the airport in a hurry.")
- 朝食を食べて、急いで学校に行った。("Having finished breakfast, I hurried to school.")
Both はやく and 急いで can be used in orders and commands,
- 急いで準備をしなさい 。("Get ready quickly!" or "Hurry up and get ready!")
- はやくしなさい。（Be quick! (i.e. Do it quickly)）
I hope this helps!