Ciao Letizia,
Before I answer I should say that my knowledge is actually from Dutch / Afrikaans, but basically the words are the same so I imagine the meanings are the same.
Rühren is, as I'm sure you realise, the basic form.
Umrühren is similar - stirring always means stirring around! In a recipe it's used where there is a bit more emphasis on the action itself, rather than on the substance being stirred. For example, it can be used to denote stirring during a long period of cooking (eg. "leave to simmer for an hour, stirring every fifteen minutes")
Einrühren: to stir in. This can also be translated simply as "add" if it's that kind of recipe (eg. for a cake)
Aufrühren I'm afraid I don't have any answer to. Hopefully a native speaker will come this way soon!