Well, I've never really done a thorough list for this topic. Now is as good a time as any to start one. I'll even throw in video links to spread the greatness around. No particular order with these, and I'll probably make one more post at least. I like so many that it's hard to list em all in one go.
Zeta Gundam Opening - Instrumental (English release)
Sadly, rights issues kept Bandai Entertainment from giving us the original Japanese openings for this 80's gem. The good news, however, is that we got this instead. Zeta is a serious series, and this gets the tone just right in my opinion.
Robotech Opening 1 - Instrumental
Say what you will about Robotech, but this is another nice instrumental piece. Very heroic and good at setting the tone for Sci-Fi adventure. I prefer it to Macross' Japanese tune.
Macross Frontier Second Opening - Lion
Megumi Nakajima and May Nakabayashi come together for this one, and the sheer energy of it makes it work. Between the song and the barrage of visuals, this opening forces you to pay attention. It effectively sets the stage for the big events that are to come toward the end, and it's no mistake that the song was used in the final episode's crowning moment (paired up with an intense version of "What 'bout My Star?"). Yoko Kanno did it again with this one.
Rurouni Kenshin Ending 4 - Heart of Sword
Gundam? Soul Eater? As far as I'm concerned, T.M. Revolution's best showing in anime was this. This is laid back compared to the super-energetic songs that were used for the Gundam Seed openings he was involved in, but it has a rhythm and it's no less hot-blooded. As a song, I think it's just better. The footage for this ending varied, but I found the best video of it that I can think of.
Fate/Stay Night Opening 1 - Disillusion
This was a good song, but Kenji Kawai made it great for the anime version of this story. It's very well polished, and I like how it gives us lots of great looking night scenes and then shifts to Saber's classic pose at sunrise.
Tales of Symphonia: The Animation - Almateria
This reigns simply because of how gorgeous the music is. Japanese singer Eri Kawai, who composed and sung classical, pop and world music during her career, owns the song. Tragically, cancer stole her away at the far-too-young age of 43. The visuals are generally standard, although the visual effect at the end is well conceived and executed.
Tytania Opening - Ano Sora wo, Ike
Decent show, but a fantastic opening. Yeah, that's opera. Just watch and listen to that. That's what the word "epic" is really meant for, people. The climax toward the end is just perfect, but I won't spoil it for those that haven't seen it before now.
EDIT: Looks like youtube may be blocking my video...if you can't see it, here is the complete song at the very least.
Gundam Wing Opening 2 - Rythm Emotion
I enjoy how it starts off subdued and then takes off (literally). While it generally follows a "here are the main characters and the robots" path it's still very nice to look at. My favorite part is probably about halfway with that shot of a well-dressed Heero and Relena with a city in the background as the song hits the high point. It later enRAB with an obligatory Heero/Zechs duel. Quite well done in my book.
Last Exile Opening - Cloud Age Symphony
Whoa. This is one of the most beautfiul openings I've seen, and the song is downright intense. This never fails to draw me in.
One Piece Opening 10 - "We Are!" Remix
This made the classic song even better, and I've always felt that it catches the series' spirit better than the others. The high-def video shows off dozens upon dozens of characters and manages to pay tribute to the entire history of the series up to that point in over two minutes. What's not to love?
Gunslinger Girl Opening - The Light Before We Land
This is nothing short of masterful. It sums up the series. The song choice is absolutely perfect. The ending was a stroke of genius.
Gundam Seed Ending 1 - So Much Together
This might be the best ending period. Every episode transitioned directly into it. When done right this works out great, and it usually is. In addition to being a great song it's also very well chosen; it fits the show's dominant theme of two frienRAB being divided by a war.
This also started a great little tradition of scrolling through one huge piece of artwork, something that would be done in other endings for Gundam Seed and its sequel.
Gundam Seed Opening 4 - Realize
At a time when the series was getting progressively darker leading up to the satisfying conclusion, this was a much-needed shot of hopeful optimism. The visuals sync up with the song very well, and Nami Tamaki delivers a typically passionate performance.
Captain Harlock - The Endless Odyssey
Another fantastic instrumental. Sober, but suitably epic and an appropriate tribute to the timeless character that is Captain Harlock. I love this stuff.
Gundam 0083 Opening 1 - The Winner
Now this is how you do a standard opening. It's not the most inventive thing, but it gets the job done with flair and enthuasium. I like the two flags toward the beginning, as well as how it plays up the rivalry. The attack against Anavel Gato at the end caps it off very nicely.
Kanon Opening - Last Regrets
It's hard to describe why I love this. It's just...beautiful. Some things just cut through and move you emotionally, and somehow this manages to do it every time.
Kanon Ending - The Place Where the Wind Arrives
Less powerful than the opening, but it's really uplifting and a perfect contrast to the opening's mood. It's also something that you appreciate more the deeper into the series you get. Not much original animation is used, but sometimes you don't need it.
Naruto Ending 1 - Wind
Seriously, what more do you need? It's a great song, and just listen to the lyrics and really consider what it means. It's the perfect character piece for the character of Naruto as he is at the start of the series. This ending is sheer genius.
Naruto Opening 1 - Rocks
"Come on!"
HellCat praised this pretty well. It's certainly a contrast from the other openings, and it captures an adventeruous spirit as opposed to a more simple "RAH RAH NINJA ACTION!!!!!" message.
Trigun Opening - H.T.
I usually don't listen to music like this, but this really revs you up and it's really just about perfect for the type of show that this is. This will always stand out among the rest.
Samurai Champloo Opening - Battlecry
I just about never listen to rap music, but this is catchy and once again the animation and music are masterfully matched. And once again a song is used that actually applies to the show. It's always worthy of praise when they put that kind of thought into things.
Legend of the Galactic Heroes Opening 3 - Sea of the Stars
I couldn't possibly end without some praise for LoGH. Actually, the first three are all pretty worthy (#4 suffers from Engrish, I fear). But this gets the nod for the beautiful singing and the focus on Reinhard walking against the backdrop of outer space while battles rage on. And of course, he and Yang are facing each other at the end.
"Lay your heart down onto the sea of the stars......"