You probably noticed that the Psalms are written in verse. If you didn't, you might want to go back for a reread. Songs and poetry appear elsewhere in the Bible, but Psalms is by far the biggest collection.
Many of them are attributed to King David (you'll see "a psalm of David" as a title a lot), an Israelite figure famed for his musical and poetic skills. Scholars are at odds on whether they were written by David, or for David, or in honor of David, but no matter what side of the debate you fall on, it's clear that David was a major formative influence. David, after all, was famously pious and spiritually developed, but also super-flawed. Classic profile of a poet, right?
What we do know is that the Psalms are written to suit a bunch of different situations. The hilariously named German scholar Hermann Gunkel (elementary school couldn't have been easy) came up with five basic subgenres for the Psalms (there are other ways to do it, but let's go with the Gunks):
Hymns: praise songs, celebrating God's supremacy or his actions in history. This is the "God, you're the best" subgenre.
Communal Laments: Everybody's sad, and everybody's complaining together. These are poems in which the community, as a whole, explains and complains that something awful has happened, and either (a) protests their innocence, (b) apologizes, or (c) asks for help...pronto.
Royal Psalms: As you could probz guess, these are Psalms that deal with royal themes, like the life and times of a human king or describing God as a king.
Individual Laments: These are like communal laments, but are usually a little more optimistic (apparently, the group drags everybody down)—individual laments are way more likely to express confidence that God will deliver the speaker from their turmoil.
Thanksgiving Psalms: This is when a speaker thanks God for something good (deliverance from suffering, Six Flags tickets, etc.)
Keep your eyes peeled for examples of all of these—they'll help you recognize how similar the Psalms can be. It's not just a diverse soup of un-datable, anonymous compositions. They have a lot in common.
Speaking of similarities, here's some literary elements to be aware of:
Repetition: We've got that in spades. Passages repeat within the Psalms themselves and even pop up later in the book. Take a look at Psalms 7:15 and 9:15, 85:9 and 85:12-13 for some good examples. Certain Psalms even rehash earlier books, like Genesis (105) and Exodus (78, 106, 135, and 136).
Parallelism: The writers of Psalms often used the same phrasing structure but with different words. So, He who planted the ear, does he not hear? He who formed the eye, does he not see? He who disciplines the nations, he who teaches knowledge to mankind, does he not chastise? (94:9-10) Packs a punch, right? Especially when these words are chanted or spoken aloud. Go ahead. Try it. We'll wait. Hungry for more parallelism? Check out 19:7-10, 20:2-5, 24:7-10, 35:5-6, 38:7, 103:19-22, 118:1-9, and 136.
Selah: This word is littered throughout Psalms. Scholars are unsure of its meaning, but it probably indicates a pause, a musical interlude, or is just another way of breaking up the text (source).