Hi there! :) well, that's maybe a tough question. In Italian we have 3 verbs to express ability: potere, riuscire, sapere.
"Potere" is used: for asking favours (Puoi aprirmi la porta? = Can you open the door for me?), asking/giving/talking about permissions ("Qui non puoi fumare = you can't smoke here; posso fumare? = can I smoke?; se vuoi, puoi fumare = if you want, you can smoke)
"Riuscire" is used: to express ability to do something (Riesco a leccarmi il gomito = I can lick my helbow - okay that was a silly example).
"Riuscire" is almost interchangeable with "sapere", but with 2 little differences: "riuscire" sometimes implies more struggle in doing something, especially if used at the past tense (sono riuscito a imparare l'italiano! = I was able to learn Italian!); and again "riuscire" is always used together with a verb, while "sapere" works with nouns (sono riuscito A IMPARARE l'italiano/io so L'ITALIANO).
And 2 last little tips: never confuse "potere" and "sapere", many learners of Italian say "io posso parlare italiano!" but that's wrong! ; "riuscire", like "potere", can be used for asking favours, but it implies that you're doubting about one's ability ("puoi aprirmi la porta?" = just a favour, "riesci ad aprirmi la porta?" = you're asking a favour doubting whether the person actually can open the door for you, maybe he's carrying heavy things and can not, or maybe the door is difficult to open).