The first video is the main theme from Princess Mononoke, The Legend of Ashitaka. The second video is the theme with lyrics (by Hayao Miyazaki) to Joe Hisaishi's music in Japanese. Yoshikazu Mera performed it. The last movie is the same theme, but in English, performed by Sasha Lazard.
Mononoke Hime
Lyrics by Hayao Miyazaki
Music by Joe Hisaishi
Japanese
Haritsumeta Yumi no Furueru Tsuru yo
Tsuki no Hikari ni Zawameku Omae no Kokoro
Togisumasareta Yaiba no Utsukushi
Sono Kissaki ni Yoku Nita Sonata no Yokogao
Kanashimi to Ikari ni Hisomu Makoto no Kokoro wo
Shiru wa Mori no Sei
Mononoke Tachi dake
Mononoke Tachi dake
English Translation (by Deborah Goldsmith and Ryoko Toyama)
The trembling string of a taut bow
Your Heart is unsettled by the Moonlight
The beauty of a sharp blade
Thy profile, very much like its edge
Thy true heart hides behind the sorrow and anger
The spirits of the forest know it
Only the Mononoke
Only the Mononoke
Thanks!! Lyrics from: http://www.nausicaa.net/wiki/Princess_Mononoke_(song)
In the moonlight I felt your heart
quiver like a bow string's pulse
in the moon's pale light
you looked at me
Nobody knows your heart
when the sun has gone I see you
beautiful and haunting but cold
like the blade of a knife so sharp so smooth
nobody knows your heart
All of your sorrow, grief and pain
locked away in the forest of the night
Your secret heart belongs to the world
of the things that sigh in the dark
of the things that cry in the dark.