There are several songs that you will acquire as you journey throughout Aquaria. You must unlock them to use them, and they give you various abilities, such as changing forms or affecting your environment. These are automatically recorded into your song menu when you first discover them, with a few exceptions.
The first song you acquire, Shield Song protects absorbs damage when hit. Be aware that this effect fizzles should you change into another form, leaving you vulnerable to damage.
The Bind Song allows you to drag certain objects, having them trail behind you as you swim. This spell is primarily used to access new areas, but can also be used offensively by dropping rocks onto enemies, or removing shells from turtles, for example.
After you meet Li, you can use Li's Song to call him to you if he has gotten stuck off screen. When he is on screen, this toggles whether or not he attacks. An easy way to remember Li's song is the way it makes an "L" shape on the song circle.
Additionally, there are a few significant songs that don't get recorded automatically when discovered:
The Ghost Mother's Song
The Ghost Mother's Song is necessary to proceed through certain portions of the Abyss and the Body. It's written on a wall deep in the bowels of The City, but in a somewhat difficult to understand format. The song is also continuously sung by the Ghost Mother herself, and can be copied from her with careful listening. Given that this song is rather complicated this can be a frustrating task for those without a well developed sense of pitch.
The Whale's Song
This song allows entry into the stomach of the whale that resides in the Abyss, to the left of the transporting sea turtle, past a sun-form door and some steam. The song can be found inscribed on a wall over an image of a whale hidden behind a false wall and a heavy current in the area between Open Waters and The Abyss, just above the Arnassi Ruins.
The Seahorse's Song
Played to start the race mini-game in the Arnassi Ruins. Unlike the others these notes are written on the wall in the only place you'd need to use them. Mentioned here only for completion's sake.