Depending on where you see this episode, it's either one long episode or two, Pt1 and Pt2. I saw it as one.
A superbeing named Q puts the Enterprise on trial for all the sins of humanity. After some clever back-and-forth with Captain Picard, Q agrees to test the Enterprise crew as the representative of humanity's current state rather than its past sins. The Enterprise passes the first test when they save an alien from enslavement at Farpoint Station.
This episode sets the tone of the whole series by asking if the mythical Star Trek future represents a better humanity. As Q points out, humans may be at peace with themselves, but they still get into deadly conflicts with other races. The Ferengi are specifically mentioned by the Farpoint leader as a competing alliance that the Federation isn't on good terms with. The problem with an alliance like the Federation is that it can't possibly fix everything about humanity. Humans will still be human, which means that there will always be evil people out there, as well as people who make egregious mistakes that produce negative outcomes regardless of good intent. The real question, then, is whether or not this is enough to condemn all of humanity, or if being good enough is good enough. Naturally, this question isn't really answered in the episode. Q complains that the Farpoint challenge was too easy, and practically promises that he'll be back in later episodes. There is hope, though, seeing as Q was ready to immediately condemn humanity at the beginning of the episode, and by the end he's backed off considerably.
One of the interesting things about the theme of the Next Generation is how well it separates the series from the Original Series, and right from the get-go. The Federation has always been presented as a politically complex entity, but this time there is the promise that this will impact every episode, and that Klingons aren't the only enemy worth counting. This complexity extends to the characters with their involved introductions and pre-existing relationships, from the romance of Troy and Riker to the more awkward I-knew-your-father-before-he-died that Picard has with Wesley. If the original series was all action with a hit-you-over-the-head morality, The Next Generation promises to keep the big questions and the action, but to diffuse them with realistic complexities.
Fans of the original series still have a lot to like in The Next Generation. There are plenty of signs in this first episode that say 'we're still Star Trek!'. The biggest nod to the original series, and the one that got a fan girl squeal from me, is the appearance of Deforest Kelley as a 137yr old Dr. McCoy. He appears on the Enterprise because - well, just because, actually. The plot barely justifies him being there except to help set the time frame for when the Next Gen occurs. His other purpose is to compare Data to Vulcans (favorably, of course) so that viewers who have reservations about seeing an android on the show can get over it. Dr. McCoy says it's ok and that 'makes it so'. No, really!
Then for people who are less of a rabid fan girl than me, there are smaller nods to the original series. For instance, Q showing up in Shakespearean clothes. Outrageously advanced aliens are always getting things like that wrong, although Q seems to do it on purpose. And the 21st century court that Q takes them to? Star Trek loves to use the formula [real history] + [real history] + [made up history] = the future. Finally, Farpoint is yet another take on the too-perfect-to-be-true settlement. If it's not run by aliens who want to trap humans, it's powered by an alien that's been trapped by humans.
Such tropes are comforting, but in the end it's not them, nor the theme, that has me geared up for the next episode. It's the characters. Picard is a strong-willed captain who's afraid he'll make an ass of himself in front of children. Wesley's a kid who wants to see the bridge just because it's the bridge, dammit. La Forge would rather see with pain than not see at all, and Riker is our go-to dude for the dangeresque. I think I can even put up with Troi...