연애 literally means 'be love with someone' but it is simply a noun form of 'seeing someone) and it almost always goes with the verb '하다'.
연애하다 is same as ~와/랑 사귀다.
If you watched many K-dramas, you may have noticed that a lot of their titles include this word 연애. (For example, 연애시대, 연애말고 결혼).
1) 요즘 나 연애해. I'm seeing someone these days. I'm in love.
2) 너 요즘 연애 안하니? Aren't you seeing someone lately?
3) 나 지금 연애 안해. I'm not seeing anyone right now.
For other vocabulary such as '돈 독이 들었나?', '그게 사랑 아니냐?', please check http://www.laughingbeans.com/learn/class/377.
These vocabularies come from a Korean movie called 남자가 사랑할 때 (Man in Love).
The movie, which was released in 2014, features one of the greatest actors in Korea, 황정민 (Jungmin Hwang). This is love story between a loan collector working for loan sharks and a daughter of a debtor that he was blackmailing to collect the loan. Quite a typical storyline, but Jungmin's acting just brought this movie to another level.
If you are new to Korean movies, I highly recommend you watch some movies that featured him. https://en.wikipedia.org/wiki/Hwang_Jung-min