When talking about classical Japanese literature, it is almost impossible to forgo mentioning certain genres. This list is introductory and rough at best but hopefully still useful for someone only getting into Japanese classics.
Waka 和歌 or Japanese poems, are most often short poems of 31 syllables, written in the pattern of 5-7-5-7-7. They are also called tanka 短歌 or short poems, but this term mainly refers to modern compositions in the format. Waka also encompasses long poems or chōka 長歌, written in a pattern 5-7-5-7...5-7-7.
In classical Japanese literature, poetry has long been the genre of paramount importance and played a significant role in social relationships, although by 12th or 13th century most of the poetry came to be composed for poetry contests or utaawase 歌合 or/and in sequences of 100 poems, called hyakushu 百首. A good introduction to Japanese poetry is a collection of one hundred poems called Ogura Hyakunin Isshu 小倉百人一首 (Mount Ogura’s One Hundred Poems by One Hundred Poets), most of the posts on this blog are dedicated to.
Monogatari 物語 or tales encompass a wide variery of Japanese vernacular prose. Although monogatari are mostly prose, they frequently also include poetic compositions. Monogatari can greatly vary in lenght and may or may not be based on real events. Genji monogatari 源氏物語 or The Tale of Genji (early 11th century; see excerpts) is the most famous work of the genre.
Nikki 日記 means "diary", but the genre also includes many works that are closer to memoirs than diaries. Men’s diaries, with a few exceptions, were day-to-day records written in Chinese, while women’s diaries were written in Japanese and are more often close to memoirs. Diaries also often include poems, as poetry was an inseprable part of social life. Among men’s diaries in Chinese, Fujiwara no Teika’s Meigetsuki 明月記 (Record of Clear Moon) will be one often mentioned here, and as for diaries in Japanese — Ki no Tsurayuki’s Tosa Nikki 土佐日記 (Tosa Diary) is particularly famous. Among women’s diaries, Murasaki Shikibu Nikki 紫式部日記 (Diary of Murasaki Shikibu) has already been mentioned on this blog.
Zuihitsu 随筆 is a genre close to essays, although not exactly that. Mostly prose, once again, works of this genre can contain poetry and a particularly famous work is Sei Shōnagon’s Makura no Sōshi 枕草子 (The Pillow Book).