The first is from Heroes of Mana, a DS real-time strategy game from the Seiken Densetsu/Mana series. I've never played it, but I looked up the soundtrack as soon as I heard that Yoko Shimomura composed it. The track is cheerful and bouncy. I've only seen the title translated as "Tenacity," so I included this title in the update, but the original Japanese title is 頑張りやさん (ganbariya-san), where ganbariya appears to mean something like "hard worker" or maybe even "overachiever." I listed the title as "Mr. Hard Worker." If a native Japanese speaker could shed some light on this, I would really appreciate it.
The other piece, from Seiken Densetsu 3, is quite literally called "Ancient Dolphin." [The SD3 OST's track names are almost entirely katakana spelling out English words--in this case: エンシェント・ドルフィン enshiento dorufin]
This piece is quiet and mysterious. Edited 9/15/11: