Lyrics can be the words to a song, sure, but the word lyric can also refer to a kind of poetry. Lyric poetry is all about giving us a glimpse inside the speaker's head. That means lyric poetry is usually written from the first-person point of view (using the pronoun "I") in order to directly convey the speaker's thoughts and emotions.
Wait, so why in the world are they called lyrics? Can't they get their own term?
Lyric poetry includes lots of subgenres like the ode and the sonnet. Check out our analysis of Elizabeth Barrett Browning's "How do I love thee?" or Percy Shelley's "Ode to the West Wind" for some classic examples of lyric poetry. No lyre necessary.