あたし is the same as 私, but more feminine, I guess. Girls say it alot. I kindof got in the habit of throwing it into my (crappy) Japanese because I read so much shoujo manga and listen to girl music and watch shoujo anime.

あたしの世界 would mean more like "my world" as the の shows posession to the proceeding word, and the word after word is the word that is posessed.

I really should change that, though, because あたしの世界 doesn't really fit my character at all anymore. I've completely stopped using atashi informally and usually use boku or ore now. I've gotten over my Japanese girly-talk stage XD