What is a poem? This is not as simple a question as it seems. English language poetry used to be the kind of thing people read to each other after dinner but with the advent of radio and television and the internet, popular consumption of poetry has been radically limited. And yet, there is a modern distinction between poems and songs that would be irrational in the ancient world. The great epic poem, the Odyssey, was sung for hundreds of years before it was written down. Ancient poetry, as well, focused on different devices than English poetry. For instance, instead of emphasizing rhyme and meter, Hebrew poetry emphasizes parallelism. For instance, in Psalms 29:1 

Give unto the LORD, O ye mighty,
give unto the LORD glory and strength.

The last word of each line parallels each other. 

Another great poetic example is the entirety of Psalm 119. The longest chapter in the bible is an acrostic of the Hebrew Alphabet all the way through. 

The poetical books of the Bible are generally considered to be Job, Psalms, Proverbs, Ecclesiastes, and the Song of Solomon. I would be put Lamentations in there, but not everyone does. 

What is important is that these books were not written to relay information. They had a different purpose. In the early church, leaders would teach people first Ecclesiastes because the hard look at life was the most similar to what everyday people experienced. Then, they would teach the Proverbs showing the way of wisdom. And finally, after years, they would allow people to read the Song of Solomon. The idea being that you must be mature in your faith to understand the Song and not be moved to temptation by the fleshiness of it. 

We have in the Psalms, as well, the book mostly heavily cited by Jesus Christ in the Gospels. It is also referred to as the songbook of the church. In Psalm 12:6, it says “The words of the LORD are pure words, like silver refined in a furnace on the ground, purified seven times.” This the basis of the ancient Liturgy of the Hours. The 7 prayer times taken up by monks and nuns in the early church. My own prayer life is heavily shaped by this practice.

The poetry books of the Bible are still Holy Scripture but they do not cease to be poems. They are meant to be spoken and shared, to evoke and provoke, to guide and emote. If you read the Psalms like you read Genesis or Judges, you are missing the point. Yet God is found therein and God speaks to us all from the pages of these poems.