Being a combination of the substitute noun それ, な, and のに, the particle meaning ‘although’ or ‘despite’, それなのに acts as a conjunction meaning ‘and yet’, or ‘despite that’, and simply refers back to the previous statement made, when the following statement is in contradiction to it.
Due to its use as a conjunction, それなのに will primarily appear at the beginning of a new sentence.

• 妻（つま）は今（いま）、風邪（かぜ）をひいて寝込（ねこ）んでいる。それなのに、いきなり動物園（どうぶつえん）に行（い）きたいと言（い）ってきた。
  MY wife is currently in bed with a cold. And yet, she suddenly told me that she wants to go to the zoo.
• 私（わたし）は毎日（まいにち）３時間（じかん）日本語（にほんご）の勉強（べんきょう）をしている。それなのに全然（ぜんぜん）上達（じょうたつ）している感（かん）じがしない。
  I study Japanese for three hours every day. And yet I don't feel like I'm improving at all.

Being that the (B) sentence is in contradiction to the (A) sentence, それなのに will often indicate something that is unexpected, or was somehow unpredictable based on (A).