Well 멋지다 (멋져) means marvelous, splendid, cool, etc. 떴다 dictionary form is 뜨다 which means float or rise(as in the sun, moon), to come up, to open one's eyes etc. It really all depends on context which you didn't leave much of. Based on the times I've come across it it usually means to rise/to come up. Hope this helps.

Hi, 예림~^^* Like on other comments it means a few things. As you said it's used when something is cool .. it should be this : when someone you are waiting for comes out. 지금 방탄소년단이 인천공항에 떴대~ so BTS is at ICA now! (so there they come!) : when something/someone is very popular. 전지현이 영화 '엽기적인 그녀'로 떴어. 전지현 became so popular because of the movie, 엽기적인 그녀. OR it also uses in.. : when someone run away/ leave or..even die 이미 그 여자 뜨고 없어요. The woman has already run away. 그 사람 이 세상 뜬지 오래됐어. so it's been quite long since he died. :-) Hope this is helpful, too! G00D Luck! xD

So in the dictionary 떴다 - 뜨다, means to float, so for example you could say, 배가 물 위에 떠 있다. The boat is floating on the water. It can also mean to open 눈을 뜨다 To open one's eyes. In slang it can also mean to appear all of a sudden, so much that you're surprised. 경찰이 떴다 The police are here! 빅뱅의 신곡이 떴다! Big Bang's new song just came out! Hope that helped!

'떴다' is the past tense of '뜨다'. The vowel 'ㅡ' is dropped from the stem '뜨-', the past pre-final ending '었' is added, and the final ending '다' is added. '뜨다' means that something float upward. Other meanings of '뜨다' are derived from this original meaning. '앨범이 떴다' means 'the album came out/has came out.'