Bonus quests are quests you will be given by doing an unlisted buy swtor credits objective while already on a quest. For example, say you need to collect flowers for Juliah, that's the only objective that the quest shows you. But in Juliah's garden are giant wasps which are in the way of you collecting any flowers for her. You won't immediately be given the bonus quest for just being in the area but when you kill one of the wasps it will pop up and let you know that you are now on a bonus quest as well. Bonus quests are not required to progress the story in any way nor have I ever seen them lead to anything important.
All I have ever seen bonus quests reward was a little bit of extra xp. Depending on what the quest has you do, the amount of XP it will give you as a reward will vary. I've seen them give 1/3 the amount of the actual quest to 2x more than the actual quest. As I mentioned before, the bonus quests are never required to advance the story line so worse comes to worse if one is buggy or extremely difficult or annoying you can just skip it. Two things you should know about bonus quests first though.
First thing you should know is once you complete the mission you're on that lead you to the bonus quest, the bonus quest will disappear whether you've finished it or not. Along the same lines of that, when you finish a bonus quest there is no need to go somewhere special to turn it in. Once completed it will automatically reward you on the spot. The second thing you should know about bonus quests is how to receive them. There is no clear way to receive a bonus quest, finding one is about luck and some common sense.