では and じゃあ are both abbreviations of
それでは, and used as 接続詞（せつぞくし） (conjunctions) in Japanese. This structure may be considered as a combination of
それ, the 格助詞（かくじょし） (case marking particle) で, and the 係助詞（かかりじょし） (binding particle) は, but is commonly treated as a single word.
それでは has two primary functions. It can be used to continue in the same topic, showing that ‘(B) is a result of (A)’. Or it may be used to change the conversation topic completely, showing that ‘(A) is at a good point to become (B)’. In each of these cases, it can be translated as ‘and with that’ in English. ‘If so’, ‘well then’, and ‘in that case’ are also common translations.
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それでは、あなたは行（い）きたくないという事（こと）ですか？
In that case, you mean that you don't want to go?
では、こちらの商品（しょうひん）はどうでしょう？
Well then, how about this product over here?
用意（ようい）できた？じゃあ、5分（ふん）で出（で）よう。
Are you ready? In that case let's leave in 5 minutes.
The translation of ‘in that case’ is most commonly used when one person is making an assumption/drawing a conclusion about something that someone else has just said.
それでは,
では, and
じゃあ are also frequently used as a polite way to say ‘goodbye’. In these situations, it comes across in a similar way to ‘well…. I best be off’ in English.