I'm going to assume you read this somewhere on the internet or during a chat. 담에 is what people use that is short for 다음에 in chat or online language. 다음 means next and 에 is time/location particle.얘 which is more colloquially short for 이 애, which is short for 이 아이. 이 is like a prefix (prefix probably isn't the correct term) for this like 이것 means "this thing". 아이 means child, but is also colloquially used as person when referring to someone in similar hierarchy like friend. 같이 means together. 안 when attached in front of verbs mean not or don't. For 갈 거임, this 임 is like a speech ending, also more commonly used in chatting. 갈 거 is short for 갈거예요 or 갈것이다 which are future tense for 가다 go. To put all of this together, it means, "Next time, I'm not going with this person" Hope this helps!

It is same to "다음에는 이 아이랑 같이 안 갈 거임" It means "Next time, I would not go with him/her/this guy/this girl/this kid"