それでも is a combination of the 代名詞（だいめいし） (substitute noun) それ, and the 副助詞（ふくじょし） (adverbial particle) でも. It is often translated simply as ‘even so’, or ‘nevertheless’. Literally, it is a bit closer to ‘even with that’.
それでも is usually used between an (A) and a (B) phrase, to show that there is more information (usually surprising) to be considered.

• 今日（きょう）は雨（あめ）が降（ふ）るよ、それでも釣（つ）りに行（い）くの？
  It's going to rain today, even with that, are you still going fishing?
• 私（わたし）は殴（なぐ）られたり、蹴（け）られたりするのが嫌（きら）いです。それでも、空手（からて）が好（す）きなので辞（や）めれないです。
  I don't like getting punched or kicked, even so, I like karate so it is hard to quit.

This grammar point primarily expresses that ‘even with’ what has been said previously, it is not all there is to think about.