A persuasive speech / paper is basically you trying to persuade someone. So it might start off like
"Dear Milo,
Have you heard of the brand new *awesome* mall that opened today? I know you hate going places but it's got *amazing* new shops and clothes *to die* for!" Then list more reason but also try to include things like "If you do go.."
So a persuasive speech would be a speaker trying to convince a crowd to listen and agree with his / her ideas on something.