Wow... it's been a while since I last thought of my favorite music... I heard quite a few songs I really liked, recently (most of them japanese - anime has great songs), so it'll be hard to choose. However!
Does instrumental-only music count? If so, the CLANNAD (anime) OST is filled with beautiful songs. My favorites must be:
(Hear this one twice and you'll fall in love with it.)
To the same heights:
Shining in the sky:
There are much, much more instrumental songs from CLANNAD, but if I wrote all of them, this list wouldn't ever end. I also like Joe Satriani. For example, Crushing day:
Now, music with voice:
First, from CLANNAD again: (Sorry to look like a fanboy, but CLANNAD has such a beautiful soundtrack, you'll never forget it once you hear it once.)
Toki wo kizamo uta ("A song to pass the time"):
From Soul Eater: I wanna be:
(Watch out, don't feel tempted to hear this out loud. Also, not everyone enjoys this.)
From Tsubasa Chronicle: Blaze:
From Maburaho: Koi no Mahou:
From Vampire Knight: Futatsu no kodou to akai tsumi:
(Watch out, there's a little blood on this video. It's the official anime opening. It shouldn't be too shocking, but if you prefer another video, here it is: http://www.youtube.com/watch?v=jeGg9lZcpJE - the full music.)
From Air TV: Tori no Uta:
From Rizelmine: Hajimete Shimashou (Let's do it for the first time):
(This song is cute. However, the Anime where it appears is 18+, as there is a lot of nudity.)
And last but not least...
From CLANNAD: Dango Daikazoku (The big Dango family):
This one's my favourite song ever. It's cute, yet it holds such a metaphorical value... I almost know the song by head.
I decided to limit myself to 1 song per anime (except for CLANNAD, as it's terribly beautiful), so many of these have great songs that I didn't mention here. You may search them if you want to.
As you can see, I'm a really varied person when it comes to music. Although all songs are Japanese, they are REALLY different from one another. Please, don't think bad of the songs just because they are Japanese - they are truly beautiful if you have an open mind.