It is a noun/an adjective verb and it means "to be diligent". It is used when describing someone who spares no effort to work hard or to care for other people.
Meg and Amy are talking about their boyfriends.
- Meg : Do you talk to your boyfriend every day?
毎日彼氏と話している？ - Amy : Yes. It's been three months since we started a long-distance relationship, but he texts me every night to know if I'm doing well. He is very caring and gentle.
うん。遠距離恋愛を始めてから三カ月経つけど、毎日連絡くれるよ。すごくまめなの。 - Meg : You're lucky. My boyfriend lives nearby, but he rarely texts me.
いいな。私の彼氏なんて近所に住んでいるのに、めったに連絡くれない。
Ken and Leo are working overtime.
- Ken : It's time to go home. When are you leaving?
帰らなきゃ。レオは何時に終わる(or帰る)の？ - Leo : I'm going to leave before 11 pm so I can catch the last train.
終電には間に合うように、11時までには出るよ。 - Ken : You never cut corners at work. You're such a diligent person.
レオは全く手を抜かないよね。まめだよね(orまめに働くよね)。 - Leo : I just want this project to be successful.
ただこのプロジェクトを成功させたいだけだよ。