First of all, what are onomatopoeia?
Onomatopoeia are words used to describe and represent sounds.
Their usage is generally to portray simple ideas from “meow” for a cat, “woof for a dog. However in Japanese, these onomatopoeia can be used for more complex, broader concepts that dont necessarily represent a sound like a noisy crowd. Theses are called mimetic words, ideophones or sound symbolism.
Types of Japanese Onomatopoeia
The two main categories are Giongo and Gitaigo. Typically, Giongo are written in katakana and Gitaigo, in hiragana. (This isn’t always the case.) Katakana gives a harder tone while hiragana is softer.
Giongo are words that rememble and represent sounds, usually made by inanimate objects. ex. Rain falling
Giseigo, a subcategory of Giongo, are words that represent vocal sounds made by humans or animals. ex. Dog bark
Gitaigo are words that represent an action, motion or state. However, they are not meant to resemble sounds. ex. “Damp”
Gijougo, a subcategory of Gitaigo, are words that have to do with emotions and feelings (psychological or bodily feelings). ex. Throbbing, grinding pain
Giyougo, sometimes forgotten, are words that describe motions and movement. ex. Moving slowly but steadily
Japanese Onomatopoeia are quite different than that of English
Writing Onomatopoeia
Technically you are supposed to write 擬音語 (ぎおんご = giongo) onomatopoeia in katakana and 擬態語 (ぎたいご = gitaigo ) mimetic words in hiragana. But this rule isn’t enforced primarily based on the writer’s preference.
There are some instances of Kanji being used although it is uncommon in everyday text.
Vocabulary
= Pseudo/mimetic = Language = Voice
= Sound = Condition = Form = Feelings
The 3 Forms of Onomatopoeia
The way you use onomatopoeia/mimetic words are slightly different depending on whether they are used as adjectives/adverbs or verbs.
Double - e.g. にこにこ
For the double form, it is usually used as an adjective.
TO-ending - e.g. にこっと
For TO form, it is mostly used as an adverb to verbs that come after.
RI-ending - e.g. にこり or にっこり
The onomatopoeia form ending in conveys a feeling of softness or slowness. Its basically the opposite of a glottal stop. Its something long, or deliberate, not short or abrupt.
+RI-ending
N-ending - e.g. ごほん

Sources

Mimetic Words - Japanese with AnimeTranslating Japanese onomatopoeia and mimetic words - HIROKO INOSE