Ninguém is very simple. It corresponds 100 per cent to the English nobody. It can be used only as a pronoun and refers exclusively to human beings.
Nenhum is normally very simple as well, but not in English. If the sentence is negative, we use any.
Ela não quer ver ninguém.
She doesn't want to speek with anyobody.
O tempo não espera ninguém.
Time doesn't wait for anybody.
Se o imoral não fosse atrativo, ninguém faria algo errado.
If immorality were not so attractive, nobody would do something wrong.
Nenhum is adjective and pronoun. It complies in gender and number with the noun it refers to or stands for. (At least in theory: If no apple is left no apples are left as well. However nenhumas maçãs, no apples, is theoretically possible and can be appear in practice, because language, as already said very often, has nothing, absolutly nothing to do with logic.)
Portuguese native speakers very often argues that the plural forms of nenhum doesn't exist, although they are mentioned in grammar books. From a logical point a plural of nenhum doesn't make any sense. If no apple is left in singular, no apples are left in plural as well. If there is not even one apple, several apples are not there as well.
The truth is, they are rarely used and they don't make sense from a logical point of view, but there is a plural form. Not only in portuguese, but in other roman languages as well. (In English however it is true: I don't eat any apple. <=> ~ I don't eat any apples.)
The sample below is from Assis de Machado, whose novel Dom Casmurro belongs to the Brasilian literature canon and is studied in all Brasilian schools. If Assis de Machado uses a plural form, there is no doubt that it exists.
Conselhos não lhe dou nenhuns.
I don't give any advice.
Furtermore the plural of nenhum is needed it the predicative noun refers to a noun in plural. (See the last example in the table.)
nenhum Singular, maskulinum adjective
Há muitos cozinheiros e nenhum chefe.
There are a lot of cooks but no chief.
Promete que não fazes nenhum disparate.
Promise that you won't do something silly.
nenhuma singular, feminine adjective
Não vou tolerar nenhuma dessas mentiras.
I won't tolerate any of these lies.
Não conhecemos nenhuma outra, não conhecemos nenhuma melhor.
We don't know any other, we don't know any better.
Não tive dúvida nenhuma.*
There was no doubt.
nenhuns Plural, masculinum adjective
Os meus amigos não são nenhuns delinquentes.
My friends are not criminals.
Sem amor, nenhuns olhos são videntes.
There are no seing eyes without love.
* nenhum can be put after the noun to stress the importance.
Nenhum trabalha na mesma empresa.
No one works in the same company.
Nenhuma é mais rápida.
No one is faster.
There is a difference between nenhuma and nem uma. Nenhuma, as a pronoun, no one , nem uma means not even one. There is a difference in the pronunciation.