According to baseball-almanac.com, Casey at the Bat is the most iconic poem about the game of baseball ever written. Written by Ernest Thayer in 1888, this poem showcases some pretty classical poetic form and observable literary devices.
The first literary thing that I notice is a set form and meter to the poem. It is written in quatrains (four verse stanzas) with an AABB rhyming pattern. This means that the first two lines rhyme with one another, and the second two rhyme with each other. That pattern is repeated throughout. Consider the example below:
The Outlook wasn't brilliant for the Mudville nine that day:
The score stood four to two, with but one inning more to play.
And then when Cooney died at first, and Barrows did the same,
A sickly silence fell upon the patrons of the game.
The meter of the poem is also set--it's iambic heptameter. This is a fancy term that essentially explains the repetitive rhythm of each line. If you notice, there's a repeated rhythm--duh DUH duh DUH duh DUH du DUH... That "duh DUH" is one unstressed syllable and one stressed syllable. It's called an iamb. (Think of the eNotes example from the reference on iamb: the word "attempt." You say aTTEMPT, not ATtempt. See?) Okay, so we've got iamb. There are seven of these in each line. That's where "heptameter" comes in. It means seven. Seven iambs per line, every time!
Another literary device that stands out to me is hyperbole, which is "obvious and deliberate exaggeration or an extravagant statement," according to the eNotes Guide to Literary Terms. You'll notice that it seems like Mudville's entire existence hangs in the balance of this baseball game. Their hope for the future or ultimate despair are described, and seem contingent on what the players do. For example, a "sickly silence" falls upon the audience, and some leave in "deep despair." Those expecting a hit from Casey cling "to that hope which springs eternal in the human breast." The poem even states that it was "likely" that the crowd would have killed the umpire if Casey hadn't discouraged it! Now, people get worked up over sporting events. But not THAT worked up. It's an exaggeration, aka hyperbole.
Finally, repetition is pretty apparent, too. Again and again we hear variations of the phrase "Casey at the bat." This achieves a couple things. First, it creates a rhythmic, pleasing sound to the poem. Second, it reiterates the fact that Casey is at the bat--Casey, the potential savior of the team, the man to depend on. This makes it all the more ironic when Casey strikes out and loses the game. (Irony--there's another one!)