だろう is used when you are making a guess or inference based on some information you have. It is softer than saying something is a fact. For example, if you see dark clouds, you might say 雨が降るだろう (It will probably rain). It corresponds to 'probably' or 'I think' in English.

In casual conversation, だろう is often shortened to だろ, but as a learner, using だろう is perfectly fine and not too formal. When you want to confirm something with someone, you can use だろう with a rising intonation, like 明日は休みだろう？ (Tomorrow is a holiday, right?). This is common in spoken Japanese.

Note that だろう is not used for polite speech; in polite contexts, use でしょう instead. For example, with a boss or a customer, you would say 雨が降るでしょう.