In the old-time stage, the limelight was a special lighting device that used a cylinder of lime to provide a very strong light. The star of a play or performance would have this strongest light pointed to them, making them stand out on the stage and be the center of attention. Someone who likes having attention payed to them is said to "enjoy the limelight". A "showoff" is any person that does and says things especially to attract attention.

fdmaxey's answer is excellent. But yes, "enjoying the limelight" means they like having everyone's attention, it makes them feel special. Show off - is usually a term used by somewhat jealous people, or it's used in a none flattering way (most of the time). For example, imagine there were 13 friends or colleagues in a room, and one could speak Spanish, and the other 12 would like to but could not, when the Spanish speaker showed everyone how good they were at speaking Spanish, everyone else would think that one person was a "show off". English people tend to think that when someone is better than them at something, they should keep it to themselves hahaha We're strange.

I believe "enjoys the limelight" is describing a woman who wants to receive the attention of other people. The "show-off" part of the sentence refers to her desire to show that she is the best/ better than others.