Why I picked each one:
*The City On The Edge Of Forever- Star Trek: The Original Series
This episode is so classic I don't think I NEED to give a reason for it...it's probably the best episode of TOS and likely the best of all the Trek episodes (all 700+ of them...WOW, that many...) If you haven't seen it, I won't spoil a thing--go. YouTube. NOW.
*Amok Time- Star Trek: The Original Series
There are three general kinds of Trek episodes, especially with TOS: shows that are meaningful and have a powerful message or a powerful character-building moment or a theme about some of the issues mankind faces and the human condition, shows that play up the sci-fi thing for fun and some comedy, and shows that focus on adventure and action; whether it's good or bad usally a Trek episode will fall into one of these three categories. "The City On The Edge Of Forever" is one of the first kind, the "message" or "theme/character" episodes, and this is the second. Why is this a comedic episode? It's just jokes and inuendos and sarcasm from start to finish, Spock firmly stating he needs to return home because of "VULCAN BIOLOGY" and Chekov's great line after the Enterprise has been turned around about four times "I think I'm going to get spacesick." If it sounds campy, it is--but for 1960s Star Trek, it works perfectly that way, and as this is one of the few episodes nearly all the Main Seven are in (only Scotty's missing and he's mentioned by name, just never appears, so at least all Seven are mentioned) it's one of the best examples of this show. Why else? Six words: Spock gets horny and fights Kirk. Enough said.
*The Enterprise Incident: Star Trek: The Original Series
The third type of Trek show, the adventure-based one. The Enterprsie in a standoff with three Romulan Warbirds as Kirk and Spock, through a ruse, attempt to steal the enemy's cloaking device as it'll make their technology so far ahead they'll likely start a war and hurt the Federation (and the Romulans here are a mix between the Roman Republic and the more facist Empire, so I guess we can still call the Enterprise crew the good guys even if their motives are politically based.) There's an unwritten rule for the adventure episodes, though--someone has to get it on with the Woman of the Week, so to speak (this WAS the 1960s...) Uusually it's Kirk who gets the action, of course, and very occasionally it'll be Scotty or Chekov or even Bones, but would you believe here it's SPOCK who gets the girl? Yep, this episode gives us a Spock who's not horny, like above, but cooly logical as always...and THAT'S what turns on the Romulan captain? OK, why not...fun episode, keeps you goijg start to end, if you ever want an hour of just fun sci-fi action that's NOT stupid or a CGI mess, this is it.
*The Best of Both Worlds Parts I and II (Star Trek: The Next Generation)
This episode is probably TNG's greatest and one of the very few that can challenge "The City On The Edge Of Forever" for the best all-time Star Trek show. If you haven't seen it--it's all there, all 10 parts over the 2-part episode...get thee to it...
*Darmok (Star Trek: The Next Generation)
I love this episode...Picard is taken down to an alien planet where there's an energy monster in a modern re-telling of The Epic of Gilgamesh, with the added plot issue that for once the crew of the Starship Enterprise can't udnerstand the alien language...they hear the words, but idon't know what they mean, as the race communicates referencing various mythological and literary moments from their history (picture taking "Juliet at her balcony" to mean you're in love and waiting for someone or "Luke, his father revealed" to mean a moment of intense personal anguish and you get the idea.) The episode's not about fighting, really, but that first kind of Trek, the meaning one, and really speaks to how we all, regardless of race, sex or religion DO share some common stories, or at least stories that are alike...no matter the religion or lack thereof we all ahve literary heroes who have some traits--courage, compassion, selflessness, a desire to grow--that are universal, and that's the great message of the episode, brought home when Picard, at the request of the dying alien captain's request, tells one of our stories from Earth...and wouldn't you know it, he picks The Epic of Gilgamesh. If you love literature or classic stories at all you'll love this episode, doubly so if you like the idea that we all have a common bond, at least, in our shared stories.
*All Good Things... (Star Trek: The Next Generation)
Simply a beautiful and fitting end to the series...I won't give anything away, but suffice it to say it's an episode that's really good when you're about to go through a big change in your life with your friends; the night before my high school graduation I picked this one out and watched it because I knew it's just the right sort of episode for that, how it shows not only how important friendships are, but both how fragile they can be and break if not cared for and yet how enduring they can be if it's with the right people...just like Picard in the show, I had a FLOOD of memories with my theatre friends (all of us sort of like the Crew of the Enterprise...if they all were as dramatic and full of in-jokes and sheer fun as Patrick Sterwart and William Shatner), all the places we'd been, where we were going...and like Picard sees, how much we really HAD grown from those little worms we were as freshman (well, I at least was a worm...) Watch it whether you like Trek or not, it's the kind of episode (the series finale of TNG) that's like the best kind of memory--not so sappy it's overdone, but just enough so you remember way back wehn, and where you are now...and smile.
*Three Stories (House M.D.)
A great episode...well, in my opinion, MOST episode of House M.D. that aren't in Season 5 are good-great, but this one's definitely a great one, with the bonus of giving us not one medical mystery but three, a whole bunch of Med-school kids for House to screw with, and the answer to the question everyone wanted to know since Day 1--what happened to House's leg that caused him to have to use the now-famous House cane?
*House's Head/Wilson's Heart (2-parter, House M.D.)
If you watch just this episode and other ones of House M.D....I'd ask you WHY, but aside from that I'd say you've picked the one that you'll probably get a LOT out of both mentally and emotionally, and for a drama that often IS more a mental and philosophical drama than an emotional one, what with House being House and the curmudgeon that he is, that's saying a lot. I didn't cry, enver do...but I know most folks do cry at this episode, and for a GOOD reason, one I won't dare spoil. If you HAVE watched House up to this point, everything pays off ten times as much and the impact is that much stronger. The whole piece is just so perfectly constructed it's amazing, even down to the title's for the episodes--"House's HEAD" and "Wilson's HEART" indeed...
*Son Of Coma Guy (House M.D.)
Of all the episodes on this list this is probably the one that's here the most for a reason I just can't quite put my finger on. It's just...good. It's...it's just what you'd expect from House, and yet...not. In terms of that Holmes/Watson dynamic House and Watson have, this is one of the best episodes, especially for the ligher side of their characters (the 2-parter above gives maybe the best dramatic acting.) And yet there are BETTER House/Wilson episodes that are fun...so why this? Well, the patient's interesting, but not nearly as much as some of the other ones are...so why this one? Chase, Cameron and Foreman all hit their marks perfectly...so why this one? Cuddy's...Cuddy, and you know what I'm going to say next, so moving on...I think it must be the guest star. Coma Guy had been referenced/"on" the show before (if you can call House usurping a coma patient's room for the cable TV and quiet while he remains motionless "on the show") but here he is...and he's a pretty damn good character, and it only gets better--24 hours until this man, who's been in a coma for over a decade, turns back into a vegetable, and House needs to question him for background information to try and diagnose his dying son, but Coma Guy, recognizing this is probably his last day to be alive and conscious on earth, wants to have fun, not answer medical queries. The solution is a GREAT House/Wilson/Coma Guy road trip...it's just perfect, culminating in the moment where Coma Guy asks ANOTHER huge question we've always wanted to know about House, "Why did YOU want tot become a doctor?" After all, House isn't a bleeding heart, sometimes we wonder if he hasn't traded that organ in for extra cranial capacity...why would someone who doesn't like most people he meets and really does take that Sartre saying "Hell is other people" as somewhat literal want to be a doctor, a profession that cries out for the opposite? House's answer adds a lot to the character, and like the other great ones here, I won't spoil the ending--go find this one on DVD (Season 3, maybe the best episode of House M.D. all told) and give it a watch.
*The Empty House (The Granada Hill Sherlock Holmes Series with Jeremy Brett)
One of my personal icons in one of his best adventures played by THE best man to ever play him with one of the all-time best Watsons (Burke was good too) and really the only Mrs. Hudson that was ever allowed to become a fixture in the show or film or film series with a great Moriarty--it's only elementary I'd pick this...