1. to have a strong obsession for somebody or something, or to desire something strongly. In the same meaning is used as well the reflexive verb "inscimmiarsi" (something that could be literally translated as "to be 'inmonkeyed'"). In particular, the expression is used in the drug-addicteds slang, meaning to be addicted to a drug or to have withdrawal symptoms for some drug.
2. To be in a bad mood or, in general, to be in a strange or particular but elusive mood
translation: lit. "I've got the monkey for this website."
meaning: "I'm obsessed with this website"
"Ha una bella scimmia per quel tipo"
translation: lit. "She's got a very good monkey for that guy"
meaning: "She really has a strong crush on that guy"
2. "Perchè sei così antipatico oggi? Cos'è, c'hai la scimmia?"
translation: lit. "Why are you so unpleasant today? What's up, have you got the monkey?"
meaning: "Why are you so unpleasant today? Are you in a bad mood?"
"Oggi Antonio è pazzo. Deve avere la scimmia."
translation: lit. "Antonio's mad today. He must have got the monkey."
meaning: "Antonio's mad today. He must be in a very unusual mood/ There's something strange (or wrong) in him/(or also) It must be happened something to him."