Anything from Angel Beats, Fate/Stay Night, Fate/Zero, and Guilty Crown.
Karma - Kokia. Phantom Requiem for the Phantom.
Lilium. Elfen lied.
Resonance - T.M.Revolution. Soul Eater.
Paper Moon. Soul Eater.

if we're talking about songs that were played by a band in the show, then..

Grl DeMo - Alchemy
ENOZ - God Knows (classic. the song that made me seek out more j-rock)
After School Tea Time - Don't Say Lazy (though the niconico band edition blows the k-on version out of the water hardcore. http://youtu.be/c6V5AiKnto8)

Anything by Yoko Kanno or Yuki Kajura is a safe bet, and most Gundam series have a few good tracks. The Big O has some nice ones as well. Although it hasn't been released separately yet, the music for Blast of Tempest has been absolutely phenomenal.

My Favorite anime songs

-Flow by Sign
-Rain by SiD
-Haruka Kanata by unlimits
-Can Do by GranRodeo
-Alones by Aqua Timez
-The hero without a name by An Cafe
-Velvet underworld by weiss kreuz
-Kings by Angela
-Let it all out by Miho Fukuhara
-The everlasting Guilty Crown by Egoist
-sakura mitsutsuki by Spyair

That's All ^_^

Elfen Lied had really good opening music. Very Haunting

Opening for Mirai Nikki, H.O.T.D, Deadman Wonderland and Clannad After Story are also great.

Sword Art Online music./...

My favorite specific songs would be:
Akatsuki no Kuruma from Mobile Suit Gundam SEED (actually my overall favorite song)
Ashitaka and San from the movie Princess Mononoke
Tank! and Real Folk Blues from Cowboy Bebop

The anime I think had the best soundtracks would be:
Cowboy Bebop and everything else Yoko Kano directed
Sketchbook ~full colors~
Clannad
Angel Beats
Mobile Suit Gundam Seed

I am rather fond of music, and the anime that have soundtracks I like the most then to be the in my top favorites.

Bleach OSTs
Katekyo Hitman Reborn OSTs
Blue Exorcist OSTs
Darker than Black (all serials) OSTs

UVERworld, Abingdon Boys School, Rookiez is Punk'd, LM.C, Monkey Majik, and ONE OK ROCK
Maaya Sakamoto, Azu, Lisa Komine, BECCA
Yuya Matsushita, Motohiro Hata, Ono Daisuke, Kamiya Hiroshi

Maji de Watashi ni Koishinasai
Well I think the anime had the same music, I don't remember but the VN's music was god tier, the anime was garbage.

I like music of SAO... and could you help me to choose one SAO kirito prop? http://www.swordartonlinesword.com/blog/sword-art-online-kirito-cosplay-costume

I swear there is a new anime music thread every month.

Bleach Soundtracks by Shiro Sagisu

Scattered Rain from Trigun is one song that has always stuck with me.

Do yourself a favor and listen to this gem.

http://www.youtube.com/watch?v=1ayJUr_j9ck

Last Exile's "Cloud Age Symphony" by Shuntaro Okino

http://www.youtube.com/watch?v=SUjiXriGFbo

Original / Romaji Lyrics / English Translation
just keep on trying, keep on flying
I will be the light

just keep on trying, keep on flying
I will be the light

Yoake wo hitotsu no hajimari to suru nara
Ima ga kita beki toki da to suru nara

if dawn is taken to be a single beginning
if now is taken to be a time that is supposed to come

Me wo aketeirareru to omou
Itsuka kimi ni mita kibou ni nita shou e

I think that my eyes can remain open
toward the chapter which resembled the hope once seen in you

Mayoi wo hitotsu no kagayaki to suru ni wa
Ten to ten wo tsunaide ten takaku tatsu ni wa

making hesitation into a single glimmer
connecting dot to dot and rising high in the sky

Kimi to no Cloud Age Symphony
Kumo no umi no mukou ni hikari ga sasu hou e

it's a Cloud Age Symphony with you
toward the direction the light shines beyond the sea of clouds

Ah
Mahou no you na isshun ni aeru no ka?
Tsumetai yo wo hikikae ni
Futari wa habataite
Eien wo tsukamu Sono hate ni wa..........

Ah
can we encounter a moment that's like magic?
with cold nights as an exchange
the two of us flap our wings
grasping eternity to its limit.........

Sekai wa hiroi Seikai wo hiroi Aoi sora ni mau
Hi wo matoi noboritsume fukai BEAT wo utsu

the world is wide we gather the right answers dancing in the blue sky
wrapping ourselves in fire, we climb to the top and strike a deep beat

Kotae wa Cloud Age Symphony
Kumo no umi wo kimi to hikari ga sasu hou e

the answer is a Cloud Age Symphony
toward the direction where you and the light shine on the sea of clouds

Ah
Mahou no you na isshun wo hikikae ni
Hohoemu kyou wo Te no hira ni
Ippai no yoake wo
Eien wo tsukamu Sono hate ni wa..........

Ah
with a moment that's like magic as an exchange
we grasp this smiling day in the palms of our hands
many dawns
and eternity to its limit.........

Ah
Mahou no you na isshun wo hikikae ni
Hohoemu kyou wo Te no hira ni
Ippai no yoake wo
Eien wo..........
beautiful Cloud Age Symphony

Ah
with a moment that's like magic as an exchange
this smiling day in the palms of our hands
many dawns
and eternity.........
beautiful Cloud Age Symphony

My personal favorite:

http://www.youtube.com/watch?v=HHI8TcFX2ac