Basically, Japanese Onomatopoeia can be divided into five categories:
The first two groups contain expressions that are used to describe actual sounds. However, as the kanji (for those of you who can read them) indicate, the Giseigo are only used for voice-related sounds (of animals or humans) such as ぶ―ん (buun = buzz), にゃん (nyan = meow) or うわーん！(uwaan = a child crying loudly).
Giongo on the other hand basically cover all the other sounds like ザーザー (zaa zaa = heavy rain) or めらめら (mera mera = suddenly bursting into flames).
Words contained in the third group, Gitaigo, are used to describe states or conditions. These are expressions such as がたがた (gata gata = rattling/clattering), むしむし (mushi mushi = hot and humid) or びしょびしょ (bisho bisho = soaked).
Giyougo, however, are usually used for motions or movements (often related to travelling from one place to another). Among these, you will find expressions like うろうろ (uro uro = wandering aimlessly) and グータラ (guutara = not having enough will power to do anything), which is probably the way many of us feel when having to leave our beds on Monday mornings.
The last group, Gijougo, contains words that describe certain feelings and emotions like i.e. ウキウキ (uki uki = cheerful) or うっとり (uttori = being fascinated by something beautiful).
Just in case you have been wondering, some onomatopoeia do in fact have kanji. Here are some examples:
燦々 (sansan = brilliant, shining)
齷齪 (akuseku = anxious feeling when under time pressure)
煌々 (koukou = bright and shining light)
However, these kanji will most seldom be seen in daily life as onomatopoeia are usually written in either Hiragana or Katakana.
Of course, these are just some examples. There are thousands of onomatopoeia in the Japanese language used in countless situations. Using them, you can talk about the weather, temperature, food, sickness, character traits, shapes and figures, accidents or even sports. They are therefore extremely convenient in daily life and not to be underestimated. Besides, they are very fun to learn.
Just go ahead and try!