It means “not particular”. It is used to describe situations in which something is not worth mentioning. It might be considered insensitive depending on the context or the situation. The words after "別に" are often omitted.
Ken doesn’t seem to focus on work and seems to look at something else.
- Leo: What are you looking at?
何を見ているの？ - Ken: Oh, nothing in particular. I haven't been getting enough sleep lately so I’m slow at work today.
あぁ、別に(何も見てないよ)。最近あまり寝ていないから、今日は仕事が捗らない。
Ken got movie tickets from his friend.
- Ken: I have extra tickets for the horror movie, but I’m not a fan of horror movies. Do you like them?
ホラー映画のチケットが余っているんだけど、あんまり好きじゃないんだ。興味ある？ - Leo: Well, that’s okay. Why don’t you give Mike those tickets? He is crazy about it.
んー、別に(興味ない)。マイクにあげたら？あいつ、好きだよ。 - Ken: It’s a deal! Thanks.
そうしよう！ありがとう。
Meg looks like she wants to say something to her boyfriend.
- Tom:What’s wrong?
どうした？ - Meg: Nothing much... (She noticed that his fly is undone, but doesn’t know what to say...)
別に...
Meg and her boyfriend are talking about their dinner.
- Meg: Do you want to eat out or stay in tonight?
今夜、外食する？それとも家で食べる？ - Tom: I don't mind, either is fine.
別にどっちでも構わない(orいい)よ。