There are many different types of poetry, but here are some of the most common:
Sonnet, Ballad, Haiku, Ode, Ekphrasis, Elegy, Free Verse and more
Sonnet: A poem with 14 lines, usually in a specific rhyme scheme. There are two main types of sonnets: the Italian sonnet and the English sonnet. Shakespeare was also very skilled in writing sonnets.
Ballad: A narrative poem often has a dramatic plot and a recurring refrain pattern. "The Diver" by Friedrich Schiller is a famous example of a ballad.
Haiku: A Japanese poem consisting of three lines, with the first and third lines, each having five syllables and the second line having seven syllables. Although Haiku is originally from Japan there are many Haikus in other languages. Matsuo Basho was a well-known Haiku poet.
Ode: A poem that often has a solemn or sublime mood and is usually addressed to a person, event or thing. "Three Odes to my friend" by Goethe is one example.
Ekphrasis: a poem that contains a description or interpretation of a work of art or another visual object. "In the Musee des Beaux Arts" by WH Auden is said to be one of the most influential Ekphrasis.
Elegy: A poem that usually expresses sadness or melancholy and often deals with death or defeat. A very common version of this style is the pastoral elegy, which Milton’s Lycidas and P B Shelley’s Adonaïs are well known for.
Free Verse: The free verse is a poem with no fixed form or structure. It often reflects the natural melody of speech or the poet's voice.
This list is incomplete; many other poems include epigram, limerick, villanelle, ghazal, sestine, tercine, and many more.