Art songs are songs created for performance in their own right, usually with piano accompaniment, although they can also have other types of accompaniment such as an orchestra or string quartet. Generally they are defined as having an identified author(s) and require serious voice-training. The German word for song, "Lied" (plural: "Lieder"), is used in French and English-speaking communities to refer to the serious art song, whereas in German-speaking communities the word used would be "Kunstlied" (plural: "Kunstlieder"). Art songs feature in many cultures, including but not limited to: Russian (romansy), Dutch (liedje), Italian (canzoni), French (mélodies), Scandinavian (sånger), Spanish (canciones).
Folk songs are songs of anonymous origin that are transmitted orally. They are frequently a major aspect of national or cultural identity. Art songs often become folk songs when people forget who the author was. Folk songs are also frequently transmitted nonorally, especially in the modern era.
At http://www.lieder.net/ you can find thousands of texts to classical art songs in many languages, with thousands of translations to English.
Song forms (haven't decided if these should be separate pages yet)
Aria and recitative (Opera)
Chanson
Ballad
Hymn
Mass
Oratorio
Spiritual
Carol
Lied (plural: Lieder)
Pibroch
Gregorian Chant and plainsong
Rock and Pop songs
Cantata
Madrigal
Ethnic or indigenous songs (a horrible term to use but I couldn't think of an alternative just yet. Aboriginal song is the only area I have any expertise on)