Lyric poems were initially told, or sung, to the playing of a "lyre", a musical instrument. This gave them a sing-song effect and a lyric poem is pleasing to the ear, more than a plot based story. Lyric poems do not have to rhyme, but should touch upon a regular beat, not unlike lyrics to a song today.
Recent lyric poetry deals more in emotion than thought. So a simple lyric poem could be about an emotion, such as, grief, heart ache, ecstatic love, etc.
A sonnet is a great example of a well formed lyric poem, such as this example:
"Italian Sonnet” by James DeFord, written in 1997:
Turn back the heart you've turned away
Give back your kissing breath
Leave not my love as you have left
The broken hearts of yesterday
.... etc... but you get the point.