Q appears on the Enterprise bridge as a human, stripped of all his Q powers, as punishment for his past unfair treatment of humans. Picard allows Q to join the crew, but not before throwing him in the brig. Q then helps the Enterprise save Bre'el IV from its falling moon, although Geordi does most of the real work. Aliens called the Calamarain then attack Q, endangering the Enterprise. Data intervenes to save Q, risking his own life, so Q decides to go off on his own rather than risk getting Data and the rest of the crew killed. Witnessing this act of selflessness, the Q continuum restores Q's powers. Q returns to the Enterprise briefly to thank Data for being nice to him while he was human - his gift is that Data briefly experiences laughter.
Have you noticed that a lot of planets are named with the numeral IV? Where are the first III Bre'el's, eh? I suppose we'll never know because they're somehow even less important to the episode than Bre'el IV and its looming moon of doom. Once Q shows up, we're pretty sure that somehow he - no wait, Geordi? - will straighten that whole mess out.
Silly names aside, this episode is designed to make us start taking Q more seriously. Paradoxically, it accomplishes this by having Q take humans more seriously. His empathy is faint, but it's a start, and it in turn makes him more sympathetic to us. No longer is he the PIA godling running around zapping people for the fun of it, with no internal consequences. Now, if he zaps, he might actually experience this little thing called guilt. He might have internal motivations that the Enterprise crew can understand and manipulate, which could sure come in handy when he's threatening to zap crew members to death for the fun of it. And, in the future, those crew members on his zap list probably won't be Data, because Q now has some modicum of respect for the android. Data is, after all, a better human that Q.
It was also a nice touch to see more of Guinan. We still know so little about her, but we were already introduced to the fact that she and Q despise each other, so it would have been negligent for her not to antagonize Q during his stint as a human. The fact that she does so with a fork just notched her up on the list of characters I care about. I'm sure she could do a lot more to him with her unknown superhuman powers, but instead she's just mean, like Q has been to the Enterprise crew. TAKE THAT, you... spork lover... you!
PS. The actor for the Q^2 is named Corbin, which is just cool. It sounds like my child's name without having anything to do with his name, etymologically speaking.