1. でしょう is the volitional form of です。
2. When でしょう is used in questions, it makes them sound more polite. Example: "What is it?" --> 何でしょうか。
3. でしょう can also be used to express various degrees of certainty, by means of (may). Example: "This medicine may be good" --> この薬がいいでしょう。
4. Since people tend to be more assertive in casual situations, でしょう has a much stronger flavor often sounding more like, "See, I told you so!". Example: だから、時間がないって言ったでしょう！"That's why I told you there was no time!"
5. だろう is the non-polite form of でしょう。It means essentially the same thing as でしょう except that it sounds more masculine and is used mostly by males.
6. だろう can be seen in sentences just like the one we've just mentioned. Example: この薬がいいだろう。Or "Probably sleeping already." もう寝ているだろう。Or "I think Mr. Tanaka won't come today" --> きょう田中さんは来ないだろうと思います。In such sentences like the last one, you can't used でしょう because と must be preceded by a non-polite form.
Hope this helps.