[verb (plain)] + だろう [i-adj] + だろう [na-adj] + だろう [noun] + だろう
だろう is the plain form of でしょう: a guess, "probably", "I suppose". It follows the plain form of a verb or an い-adjective, and comes straight after a noun or a な-adjective, with no だ in between.
For a guess about the past, put the plain past in front: いった行っただろう, たかかった高かっただろう, がくせい学生だっただろう.
The two mean the same; they differ in register. でしょう is polite and works anywhere. だろう is the normal form in essays, news writing and thoughts; in conversation it sounds blunt and masculine, and でしょ is the softer casual form.
These go beyond a plain guess.
With a rising intonation, だろう? asks the listener to agree: "right?", "isn't it?". The speaker is fairly sure already.
In speech a guess often ends in とおもう思う, which makes it personal and softer. The polite form is だろうと思います.