"you're gonna" is informal English. "you gonna" is both informal and incorrect. The correct form - which you should always use when writing - is " You are going to ....' or " You're going to ....' 'gonna' is what 'going to' sounds like when people are speaking quickly. It's ok to say it, but don't write it.

"You're gonna" is contracted slang for "You are going to". "You gonna" is contracted slang for "Are you going to". "You gonna" would introduce a question: "You gonna go to the movies today?" "You're gonna" would introduce a statement: "You're gonna mow the lawn today."

Just that the " 're " is sometimes dropped in colloquial speech. The meaning stays the same. A little advice: don't copy this style if you're still struggling with other parts of English.