It is an adverb that means "gradually". It is used when describing situations in which something progresses/changes slowly.
"次第に" and "だんだん" are interchangeable with "徐々に". "次第に" is a formal expression. "だんだん" is an informal expression. "徐々に" is used in both formal and informal situations.
It's been 3 months since I moved to Tokyo. I'm getting use to my new life little by little.
東京へ引っ越して3ヵ月が経ち、だんだん(or徐々に)新しい生活に慣れてきた。
東京へ引っ越して3ヵ月が経ち、だんだん(or徐々に)新しい生活に慣れてきた。
I caught a cold the other day but I'm getting better gradually.
先日風邪を引いたが、だんだん(or徐々に)具合が良くなってきた。
先日風邪を引いたが、だんだん(or徐々に)具合が良くなってきた。
Meg and her sister are shopping at a grocer store.
- Meg: I want to put cucumbers and tomatoes in the salad tonight. Can you please bring them from the vegetable section?
今夜サラダにきゅうりとトマトを入れたいの。野菜売り場から持って来てくれない？ - Ari: All right. As it gets hotter, the price of vegetables has been gradually falling so I think I can find the reasonable price.
いいよ。暑くなるにつれて、野菜の値段が徐々に(orだんだん)下がっているから、手頃な物があると思うなぁ。 - Meg: Yes, I hope so too. Oh, it is getting darker so we should run the errand early and prepare dinner!
そうだね。あっ、(空が)だんだん暗くなってきたから、早く用事を済ませて(*1)、夕食を作らなきゃ！
Meg are worried about Amy because she doesn't look well but she is working hard.
- Meg: May I help you with your tasks? I've already finished all of my tasks today.
手伝おうか？今日の仕事は終わったから。 - Amy: Thank you. I couldn’t work as I wanted to today．
ありがとう。今日は思ったより仕事がはかどらなくて(*2)。 - Meg: You've just gotten over your cold. Don't try to do too much from day one. Just ease into it.
風邪が治ったばかりでしょ。初日から頑張り過ぎないで。徐々に慣らしていってね。
(*1)用事を済ます is a common phrase. It is worth memorizing.
(*2)仕事がはかどる is used when you get more done than usual. It is often used in daily conversations.