In general, Japanese people don't assert themselves of something unless they are absolutely sure that it is correct. This accounts for the incredibly frequent use of 「～と思う」 and the various grammatical expressions used to express specific levels of certainty. We will go over these expressions starting from the less certain to the most certain.
「かもしれない」 is used to mean "maybe" or "possibly" and is less certain than the word 「多分」. It attaches to the end of a complete clause. For noun and na-adjective clauses, the declarative 「だ」 must be removed. It can also be written in kanji as 「かも知れない」 and you can treat it the same as a negative ru-verb (there is no positive equivalent) so the masu-form would become 「かもしれません」. In casual speech, it can be abbreviated to just 「かも」. There is also a very masculine version 「かもしれん」, which is simply a different type of negative verb.
「でしょう」 is used to express a level of some certainty and is close in meaning to 「多分」. Just like 「～です／～ます」, it must come at the end of a complete sentence. It does not have any other conjugations. You can also replace 「～ですか」 with 「～でしょうか」 to make the question sound slightly more polite and less assuming by adding a slight level of uncertainty.
If you want to sound really, really polite, you can even add 「～でしょうか」 to the end of a 「～ます」 ending.
The casual equivalent of 「でしょう」 is surprisingly enough 「でしょう」. However, when you are speaking in a polite manner, the 「でしょう」 is enunciated flatly while in casual speech, it has a rising intonation and can be shortened to 「でしょ」. In addition, since people tend to be more assertive in casual situations, the casual version has a much stronger flavor often sounding more like, "See, I told you so!"
A: Ah! We're going to be late!
B: That's why I told you there was no time!
A: You're going to eat from now aren't you?
B: So what if I am?
A: You're going to help me clean, right?
B: Huh? Is that so?
「だろう」 means essentially the same thing as 「でしょう」 except that it sounds more masculine and is used mostly by males.
A: Where is Alice?
B: Probably sleeping already.
A: You're going home already, right?
B: That's right.